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turn_id string | prefix list | reasoning_content string | content string | messages list | action_kind string | source string | repo string | stratum string | group string | prompt string | completion string | n_prompt_tokens int64 | n_completion_tokens int64 | n_seq_tokens int64 | weight float64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
math_prealgebra.7643fdb1.pr_13915772__d018ad85:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Suppose that $a$ is a multiple of 4 and $b$ is a mult... | We need answer math problem. Need solve step by step plain text then final boxed. Let's analyze carefully.
Problem: a multiple of 4, b multiple of 8. Which statements true?
A a+b must be even.
B a+b must be multiple of 4.
C a+b must be multiple of 8.
D a+b cannot be a multiple of 8.
Let a=4m, b=8n. Sum=4(m+2n). Alway... | Let \(a\) be a multiple of 4, so \(a = 4m\), and let \(b\) be a multiple of 8, so \(b = 8n\).
Then
\[
a+b = 4m + 8n = 4(m+2n).
\]
So \(a+b\) is always a multiple of 4, which means it is also always even. Therefore, A and B are true.
However, \(a+b\) is not necessarily a multiple of 8. For example, if \(a=4\) and \(... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Suppose that $a$ is a multiple of 4 and $b$ is a mult... | boxed | affine_math | math_prealgebra | math:0055 | math:0055 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Suppose that $a$ is a multiple of 4 and $b$ is a multiple of 8. Which of the following statements a... | We need answer math problem. Need solve step by step plain text then final boxed. Let's analyze carefully.
Problem: a multiple of 4, b multiple of 8. Which statements true?
A a+b must be even.
B a+b must be multiple of 4.
C a+b must be multiple of 8.
D a+b cannot be a multiple of 8.
Let a=4m, b=8n. Sum=4(m+2n). Alway... | 179 | 519 | 698 | 5 |
math_prealgebra.7643fdb1.pr_13915772__d018ad85:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Suppose that $a$ is a multiple of 4 and $b$ is a mult... | We need answer math problem. Need solve step by step plain text then final boxed. Let's analyze carefully.
Problem: a multiple of 4, b multiple of 8. Which statements true?
A a+b must be even.
B a+b must be multiple of 4.
C a+b must be multiple of 8.
D a+b cannot be a multiple of 8.
Let a=4m, b=8n. Sum=4(m+2n). Alway... | Let \(a\) be a multiple of 4, so \(a = 4m\), and let \(b\) be a multiple of 8, so \(b = 8n\).
Then
\[
a+b = 4m + 8n = 4(m+2n).
\]
So \(a+b\) is always a multiple of 4, which means it is also always even. Therefore, A and B are true.
However, \(a+b\) is not necessarily a multiple of 8. For example, if \(a=4\) and \(... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Suppose that $a$ is a multiple of 4 and $b$ is a mult... | boxed | affine_math | math_prealgebra | math:0055 | math:0055 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Suppose that $a$ is a multiple of 4 and $b$ is a multiple of 8. Which of the following statements a... | We need answer math problem. Need solve step by step plain text then final boxed. Let's analyze carefully.
Problem: a multiple of 4, b multiple of 8. Which statements true?
A a+b must be even.
B a+b must be multiple of 4.
C a+b must be multiple of 8.
D a+b cannot be a multiple of 8.
Let a=4m, b=8n. Sum=4(m+2n). Alway... | 179 | 519 | 698 | 5 |
math_precalculus.efa0847d.pr_11888382__bf24e049:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find a unit vector that is orthogonal to both $\\begi... | We need answer user: "Find a unit vector that is orthogonal to both (1,1,0) and (1,0,2). Solve step by step plain text, end with final answer in \boxed{} exactly one box and nothing after it."
We need produce final with reasoning plain text and boxed. Need compute cross product. v=(1,1,0), w=(1,0,2). Cross v x w = det... | To find a vector orthogonal to both given vectors, compute their cross product:
\[
\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} \times \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}
=
\begin{pmatrix}
(1)(2)-(0)(0) \\
(0)(1)-(1)(2) \\
(1)(0)-(1)(1)
\end{pmatrix}
=
\begin{pmatrix}
2 \\ -2 \\ -1
\end{pmatrix}.
\]
Now find its m... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find a unit vector that is orthogonal to both $\\begi... | boxed | affine_math | math_precalculus | math:0811 | math:0811 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Find a unit vector that is orthogonal to both $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\beg... | We need answer user: "Find a unit vector that is orthogonal to both (1,1,0) and (1,0,2). Solve step by step plain text, end with final answer in \boxed{} exactly one box and nothing after it."
We need produce final with reasoning plain text and boxed. Need compute cross product. v=(1,1,0), w=(1,0,2). Cross v x w = det... | 107 | 541 | 648 | 5 |
math_precalculus.efa0847d.pr_11888382__bf24e049:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find a unit vector that is orthogonal to both $\\begi... | We need answer user: "Find a unit vector that is orthogonal to both (1,1,0) and (1,0,2). Solve step by step plain text, end with final answer in \boxed{} exactly one box and nothing after it."
We need produce final with reasoning plain text and boxed. Need compute cross product. v=(1,1,0), w=(1,0,2). Cross v x w = det... | To find a vector orthogonal to both given vectors, compute their cross product:
\[
\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} \times \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}
=
\begin{pmatrix}
(1)(2)-(0)(0) \\
(0)(1)-(1)(2) \\
(1)(0)-(1)(1)
\end{pmatrix}
=
\begin{pmatrix}
2 \\ -2 \\ -1
\end{pmatrix}.
\]
Now find its m... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find a unit vector that is orthogonal to both $\\begi... | boxed | affine_math | math_precalculus | math:0811 | math:0811 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Find a unit vector that is orthogonal to both $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\beg... | We need answer user: "Find a unit vector that is orthogonal to both (1,1,0) and (1,0,2). Solve step by step plain text, end with final answer in \boxed{} exactly one box and nothing after it."
We need produce final with reasoning plain text and boxed. Need compute cross product. v=(1,1,0), w=(1,0,2). Cross v x w = det... | 107 | 541 | 648 | 5 |
math_precalculus.737d30b9.pr_9837492__8a4a65b8:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find all real numbers $k$ for which there exists a no... | We need answer math problem. Need solve step by step plain text, final boxed. Problem: find real k such that exists nonzero 2D vector v with A v = k v. That's eigenvalues of matrix [[1,8],[2,1]]. Compute characteristic: det(A - kI)= (1-k)^2 - 16 = k^2 -2k +1 -16 = k^2 -2k -15 = (k-5)(k+3). eigenvalues 5, -3. Solutions:... | The equation says that \(k\) is an eigenvalue of
\[
A=\begin{pmatrix}1&8\\2&1\end{pmatrix}.
\]
So we solve
\[
\det(A-kI)=0.
\]
Compute:
\[
\det\begin{pmatrix}1-k&8\\2&1-k\end{pmatrix}
=(1-k)^2-16.
\]
Thus
\[
(1-k)^2-16=0
\]
\[
(1-k-4)(1-k+4)=0
\]
\[
(k-5)(k+3)=0.
\]
So the possible values are \(k=5\) and \(k=-3\).
\[
... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find all real numbers $k$ for which there exists a no... | boxed | affine_math | math_precalculus | math:0773 | math:0773 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\mathbf{v}$ such ... | We need answer math problem. Need solve step by step plain text, final boxed. Problem: find real k such that exists nonzero 2D vector v with A v = k v. That's eigenvalues of matrix [[1,8],[2,1]]. Compute characteristic: det(A - kI)= (1-k)^2 - 16 = k^2 -2k +1 -16 = k^2 -2k -15 = (k-5)(k+3). eigenvalues 5, -3. Solutions:... | 132 | 352 | 484 | 5 |
math_precalculus.737d30b9.pr_9837492__8a4a65b8:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find all real numbers $k$ for which there exists a no... | We need answer math problem. Need solve step by step plain text, final boxed. Problem: find real k such that exists nonzero 2D vector v with A v = k v. That's eigenvalues of matrix [[1,8],[2,1]]. Compute characteristic: det(A - kI)= (1-k)^2 - 16 = k^2 -2k +1 -16 = k^2 -2k -15 = (k-5)(k+3). eigenvalues 5, -3. Solutions:... | The equation says that \(k\) is an eigenvalue of
\[
A=\begin{pmatrix}1&8\\2&1\end{pmatrix}.
\]
So we solve
\[
\det(A-kI)=0.
\]
Compute:
\[
\det\begin{pmatrix}1-k&8\\2&1-k\end{pmatrix}
=(1-k)^2-16.
\]
Thus
\[
(1-k)^2-16=0
\]
\[
(1-k-4)(1-k+4)=0
\]
\[
(k-5)(k+3)=0.
\]
So the possible values are \(k=5\) and \(k=-3\).
\[
... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find all real numbers $k$ for which there exists a no... | boxed | affine_math | math_precalculus | math:0773 | math:0773 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\mathbf{v}$ such ... | We need answer math problem. Need solve step by step plain text, final boxed. Problem: find real k such that exists nonzero 2D vector v with A v = k v. That's eigenvalues of matrix [[1,8],[2,1]]. Compute characteristic: det(A - kI)= (1-k)^2 - 16 = k^2 -2k +1 -16 = k^2 -2k -15 = (k-5)(k+3). eigenvalues 5, -3. Solutions:... | 132 | 352 | 484 | 5 |
math_precalculus.4a229bbb.pr_7108717__fa063b61:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find the matrix that corresponds to projecting onto t... | We need answer user: Find matrix corresponding to projecting onto vector (2,-3). Need solve step by step plain text, end with boxed. Need ensure final only one boxed block and nothing after it. Need likely orthogonal projection matrix onto span of v in R^2. v = [2, -3]^T. Projection matrix P = v v^T / (v^T v) = (1/(4+9... | To project onto the vector \(v=\begin{pmatrix}2\\-3\end{pmatrix}\), use the orthogonal projection matrix formula
\[
P=\frac{vv^T}{v^Tv}.
\]
First compute
\[
vv^T=
\begin{pmatrix}2\\-3\end{pmatrix}
\begin{pmatrix}2&-3\end{pmatrix}
=
\begin{pmatrix}
4 & -6\\
-6 & 9
\end{pmatrix}.
\]
Also,
\[
v^Tv=2^2+(-3)^2=4+9=13.
... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find the matrix that corresponds to projecting onto t... | boxed | affine_math | math_precalculus | math:0515 | math:0515 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Find the matrix that corresponds to projecting onto the vector $\begin{pmatrix} 2 \\ -3 \end{pmatri... | We need answer user: Find matrix corresponding to projecting onto vector (2,-3). Need solve step by step plain text, end with boxed. Need ensure final only one boxed block and nothing after it. Need likely orthogonal projection matrix onto span of v in R^2. v = [2, -3]^T. Projection matrix P = v v^T / (v^T v) = (1/(4+9... | 84 | 494 | 578 | 5 |
math_precalculus.4a229bbb.pr_7108717__fa063b61:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find the matrix that corresponds to projecting onto t... | We need answer user: Find matrix corresponding to projecting onto vector (2,-3). Need solve step by step plain text, end with boxed. Need ensure final only one boxed block and nothing after it. Need likely orthogonal projection matrix onto span of v in R^2. v = [2, -3]^T. Projection matrix P = v v^T / (v^T v) = (1/(4+9... | To project onto the vector \(v=\begin{pmatrix}2\\-3\end{pmatrix}\), use the orthogonal projection matrix formula
\[
P=\frac{vv^T}{v^Tv}.
\]
First compute
\[
vv^T=
\begin{pmatrix}2\\-3\end{pmatrix}
\begin{pmatrix}2&-3\end{pmatrix}
=
\begin{pmatrix}
4 & -6\\
-6 & 9
\end{pmatrix}.
\]
Also,
\[
v^Tv=2^2+(-3)^2=4+9=13.
... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find the matrix that corresponds to projecting onto t... | boxed | affine_math | math_precalculus | math:0515 | math:0515 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Find the matrix that corresponds to projecting onto the vector $\begin{pmatrix} 2 \\ -3 \end{pmatri... | We need answer user: Find matrix corresponding to projecting onto vector (2,-3). Need solve step by step plain text, end with boxed. Need ensure final only one boxed block and nothing after it. Need likely orthogonal projection matrix onto span of v in R^2. v = [2, -3]^T. Projection matrix P = v v^T / (v^T v) = (1/(4+9... | 84 | 494 | 578 | 5 |
math_precalculus.d3cc36aa.pr_3337527__42482de6:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Compute $\\tan 20^\\circ + 4 \\sin 20^\\circ.$"
}
] | We need solve problem: Compute tan 20° + 4 sin 20°. Need step-by-step plain text, final boxed. Need derive exact value likely sqrt(3)? Let's check numerically: tan20≈0.363970, sin20≈0.342020, 4sin=1.368079, sum≈1.732049 = sqrt3. So answer sqrt3. Need prove.
We need provide reasoning. Many trig identities. Need compute... | Let \(x = 20^\circ\). Then
\[
\tan 20^\circ + 4\sin 20^\circ
= \frac{\sin x}{\cos x} + 4\sin x
= \frac{\sin x + 4\sin x\cos x}{\cos x}.
\]
Using \(2\sin x\cos x = \sin 2x\), we have
\[
4\sin x\cos x = 2\sin 2x = 2\sin 40^\circ.
\]
So
\[
\tan 20^\circ + 4\sin 20^\circ
= \frac{\sin 20^\circ + 2\sin 40^\circ}{\cos 20... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Compute $\\tan 20^\\circ + 4 \\sin 20^\\circ.$"
},
... | boxed | affine_math | math_precalculus | math:0664 | math:0664 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Compute $\tan 20^\circ + 4 \sin 20^\circ.$<|im_end|>
<|im_start|>assistant
<think>
| We need solve problem: Compute tan 20° + 4 sin 20°. Need step-by-step plain text, final boxed. Need derive exact value likely sqrt(3)? Let's check numerically: tan20≈0.363970, sin20≈0.342020, 4sin=1.368079, sum≈1.732049 = sqrt3. So answer sqrt3. Need prove.
We need provide reasoning. Many trig identities. Need compute... | 75 | 1,786 | 1,861 | 5 |
math_precalculus.d3cc36aa.pr_3337527__42482de6:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Compute $\\tan 20^\\circ + 4 \\sin 20^\\circ.$"
}
] | We need solve problem: Compute tan 20° + 4 sin 20°. Need step-by-step plain text, final boxed. Need derive exact value likely sqrt(3)? Let's check numerically: tan20≈0.363970, sin20≈0.342020, 4sin=1.368079, sum≈1.732049 = sqrt3. So answer sqrt3. Need prove.
We need provide reasoning. Many trig identities. Need compute... | Let \(x = 20^\circ\). Then
\[
\tan 20^\circ + 4\sin 20^\circ
= \frac{\sin x}{\cos x} + 4\sin x
= \frac{\sin x + 4\sin x\cos x}{\cos x}.
\]
Using \(2\sin x\cos x = \sin 2x\), we have
\[
4\sin x\cos x = 2\sin 2x = 2\sin 40^\circ.
\]
So
\[
\tan 20^\circ + 4\sin 20^\circ
= \frac{\sin 20^\circ + 2\sin 40^\circ}{\cos 20... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Compute $\\tan 20^\\circ + 4 \\sin 20^\\circ.$"
},
... | boxed | affine_math | math_precalculus | math:0664 | math:0664 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Compute $\tan 20^\circ + 4 \sin 20^\circ.$<|im_end|>
<|im_start|>assistant
<think>
| We need solve problem: Compute tan 20° + 4 sin 20°. Need step-by-step plain text, final boxed. Need derive exact value likely sqrt(3)? Let's check numerically: tan20≈0.363970, sin20≈0.342020, 4sin=1.368079, sum≈1.732049 = sqrt3. So answer sqrt3. Need prove.
We need provide reasoning. Many trig identities. Need compute... | 75 | 1,786 | 1,861 | 5 |
math_intermediate_algebra.d574f56a.pr_8215337__adade5c0:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find all values of $k$ so that the domain of\n\\[b(x)... | We need solve problem. Need produce final with boxed. Let's reason carefully.
Problem: Find all values of k so that domain of b(x) = (k x^2 + 2x -5)/(-5x^2 + 2x + k) is set of all real numbers.
Domain all real means denominator never zero for any real x. Denominator D(x) = -5x^2 + 2x + k. Need D(x) ≠ 0 for all real x... | The domain is all real numbers exactly when the denominator is never zero:
\[
-5x^2+2x+k \neq 0 \quad \text{for all real } x.
\]
This quadratic has no real roots when its discriminant is negative:
\[
\Delta = 2^2 - 4(-5)(k) = 4+20k.
\]
So we need
\[
4+20k < 0
\]
\[
20k < -4
\]
\[
k < -\frac15.
\]
\[
\boxed{k<-\... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find all values of $k$ so that the domain of\n\\[b(x)... | boxed | affine_math | math_intermediate_algebra | math:0220 | math:0220 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Find all values of $k$ so that the domain of
\[b(x) = \frac{kx^2 + 2x - 5}{-5x^2 + 2x + k}\]is the ... | We need solve problem. Need produce final with boxed. Let's reason carefully.
Problem: Find all values of k so that domain of b(x) = (k x^2 + 2x -5)/(-5x^2 + 2x + k) is set of all real numbers.
Domain all real means denominator never zero for any real x. Denominator D(x) = -5x^2 + 2x + k. Need D(x) ≠ 0 for all real x... | 109 | 795 | 904 | 5 |
math_intermediate_algebra.d574f56a.pr_8215337__adade5c0:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find all values of $k$ so that the domain of\n\\[b(x)... | We need solve problem. Need produce final with boxed. Let's reason carefully.
Problem: Find all values of k so that domain of b(x) = (k x^2 + 2x -5)/(-5x^2 + 2x + k) is set of all real numbers.
Domain all real means denominator never zero for any real x. Denominator D(x) = -5x^2 + 2x + k. Need D(x) ≠ 0 for all real x... | The domain is all real numbers exactly when the denominator is never zero:
\[
-5x^2+2x+k \neq 0 \quad \text{for all real } x.
\]
This quadratic has no real roots when its discriminant is negative:
\[
\Delta = 2^2 - 4(-5)(k) = 4+20k.
\]
So we need
\[
4+20k < 0
\]
\[
20k < -4
\]
\[
k < -\frac15.
\]
\[
\boxed{k<-\... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Find all values of $k$ so that the domain of\n\\[b(x)... | boxed | affine_math | math_intermediate_algebra | math:0220 | math:0220 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Find all values of $k$ so that the domain of
\[b(x) = \frac{kx^2 + 2x - 5}{-5x^2 + 2x + k}\]is the ... | We need solve problem. Need produce final with boxed. Let's reason carefully.
Problem: Find all values of k so that domain of b(x) = (k x^2 + 2x -5)/(-5x^2 + 2x + k) is set of all real numbers.
Domain all real means denominator never zero for any real x. Denominator D(x) = -5x^2 + 2x + k. Need D(x) ≠ 0 for all real x... | 109 | 795 | 904 | 5 |
math_counting_probability.d1a825ab.pr_6373282__146baa3f:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Ryan has 3 red lava lamps and 3 blue lava lamps. He a... | We need solve probability. Need reason carefully. Problem: Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is probability that leftmost lamp is blue and off, and rightmost lamp is red and on?
We need interpret. There are 6 disti... | The arrangement and the on/off choices are independent.
For the colors:
- Probability the leftmost lamp is blue: \(\frac{3}{6}\).
- Given that, probability the rightmost lamp is red: \(\frac{3}{5}\).
So
\[
P(\text{leftmost blue, rightmost red})=\frac{3}{6}\cdot\frac{3}{5}=\frac{3}{10}.
\]
For the on/off states:
Ryan... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Ryan has 3 red lava lamps and 3 blue lava lamps. He a... | boxed | affine_math | math_counting_probability | math:0005 | math:0005 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and... | We need solve probability. Need reason carefully. Problem: Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is probability that leftmost lamp is blue and off, and rightmost lamp is red and on?
We need interpret. There are 6 disti... | 116 | 1,738 | 1,854 | 5 |
math_counting_probability.d1a825ab.pr_6373282__146baa3f:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Ryan has 3 red lava lamps and 3 blue lava lamps. He a... | We need solve probability. Need reason carefully. Problem: Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is probability that leftmost lamp is blue and off, and rightmost lamp is red and on?
We need interpret. There are 6 disti... | The arrangement and the on/off choices are independent.
For the colors:
- Probability the leftmost lamp is blue: \(\frac{3}{6}\).
- Given that, probability the rightmost lamp is red: \(\frac{3}{5}\).
So
\[
P(\text{leftmost blue, rightmost red})=\frac{3}{6}\cdot\frac{3}{5}=\frac{3}{10}.
\]
For the on/off states:
Ryan... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Ryan has 3 red lava lamps and 3 blue lava lamps. He a... | boxed | affine_math | math_counting_probability | math:0005 | math:0005 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and... | We need solve probability. Need reason carefully. Problem: Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is probability that leftmost lamp is blue and off, and rightmost lamp is red and on?
We need interpret. There are 6 disti... | 116 | 1,738 | 1,854 | 5 |
math_algebra.44961735.pr_12335724__6fd2867f:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Evaluate $\\log_82$."
}
] | We need answer user: Evaluate log_8 2. Need reason step by step plain text, then final boxed. Need be careful notation: \log_82 could mean log base 8 of 2? The TeX: $\log_82$ is ambiguous: \log_8 2? Usually log_8 2, because subscript 8 then 2. Evaluate: log base 8 of 2 = 1/3 because 8=2^3, so log_8 2 = x => 8^x=2 => (2... | Let \(x = \log_8 2\). Then \(8^x = 2\). Since \(8 = 2^3\), we have \((2^3)^x = 2\), so \(2^{3x} = 2^1\). Thus \(3x = 1\), giving \(x = \frac{1}{3}\).
\boxed{\frac{1}{3}} | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Evaluate $\\log_82$."
},
{
"role": "assistant... | boxed | affine_math | math_algebra | math:0759 | math:0759 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Evaluate $\log_82$.<|im_end|>
<|im_start|>assistant
<think>
| We need answer user: Evaluate log_8 2. Need reason step by step plain text, then final boxed. Need be careful notation: \log_82 could mean log base 8 of 2? The TeX: $\log_82$ is ambiguous: \log_8 2? Usually log_8 2, because subscript 8 then 2. Evaluate: log base 8 of 2 = 1/3 because 8=2^3, so log_8 2 = x => 8^x=2 => (2... | 63 | 248 | 311 | 5 |
math_algebra.44961735.pr_12335724__6fd2867f:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Evaluate $\\log_82$."
}
] | We need answer user: Evaluate log_8 2. Need reason step by step plain text, then final boxed. Need be careful notation: \log_82 could mean log base 8 of 2? The TeX: $\log_82$ is ambiguous: \log_8 2? Usually log_8 2, because subscript 8 then 2. Evaluate: log base 8 of 2 = 1/3 because 8=2^3, so log_8 2 = x => 8^x=2 => (2... | Let \(x = \log_8 2\). Then \(8^x = 2\). Since \(8 = 2^3\), we have \((2^3)^x = 2\), so \(2^{3x} = 2^1\). Thus \(3x = 1\), giving \(x = \frac{1}{3}\).
\boxed{\frac{1}{3}} | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Evaluate $\\log_82$."
},
{
"role": "assistant... | boxed | affine_math | math_algebra | math:0759 | math:0759 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Evaluate $\log_82$.<|im_end|>
<|im_start|>assistant
<think>
| We need answer user: Evaluate log_8 2. Need reason step by step plain text, then final boxed. Need be careful notation: \log_82 could mean log base 8 of 2? The TeX: $\log_82$ is ambiguous: \log_8 2? Usually log_8 2, because subscript 8 then 2. Evaluate: log base 8 of 2 = 1/3 because 8=2^3, so log_8 2 = x => 8^x=2 => (2... | 63 | 248 | 311 | 5 |
math_algebra.cb713dcb.pr_15852248__f65e733f:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Solve \\[\\frac{2x+4}{x^2+4x-5}=\\frac{2-x}{x-1}\\]fo... | We need solve equation. Need final with boxed. Let's analyze carefully.
Equation: (2x+4)/(x^2+4x-5) = (2-x)/(x-1).
Domain: denominators nonzero. x^2+4x-5 = (x+5)(x-1). x-1 nonzero. So x ≠ 1, -5.
Simplify LHS: (2x+4)=2(x+2). Denom (x+5)(x-1). RHS (2-x)/(x-1) = -(x-2)/(x-1).
Since x≠1, can multiply both sides by (x-1... | We need solve
\[
\frac{2x+4}{x^2+4x-5}=\frac{2-x}{x-1}.
\]
First note the restrictions from the denominators:
\[
x^2+4x-5=(x+5)(x-1),
\]
so
\[
x\neq 1,\qquad x\neq -5.
\]
Factor the left side:
\[
\frac{2x+4}{(x+5)(x-1)}=\frac{2-x}{x-1}.
\]
Since \(x\neq 1\), multiply both sides by \(x-1\):
\[
\frac{2x+4}{x+5}=... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Solve \\[\\frac{2x+4}{x^2+4x-5}=\\frac{2-x}{x-1}\\]fo... | boxed | affine_math | math_algebra | math:0515 | math:0515 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Solve \[\frac{2x+4}{x^2+4x-5}=\frac{2-x}{x-1}\]for $x$.<|im_end|>
<|im_start|>assistant
<think>
| We need solve equation. Need final with boxed. Let's analyze carefully.
Equation: (2x+4)/(x^2+4x-5) = (2-x)/(x-1).
Domain: denominators nonzero. x^2+4x-5 = (x+5)(x-1). x-1 nonzero. So x ≠ 1, -5.
Simplify LHS: (2x+4)=2(x+2). Denom (x+5)(x-1). RHS (2-x)/(x-1) = -(x-2)/(x-1).
Since x≠1, can multiply both sides by (x-1... | 91 | 860 | 951 | 5 |
math_algebra.cb713dcb.pr_15852248__f65e733f:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Solve \\[\\frac{2x+4}{x^2+4x-5}=\\frac{2-x}{x-1}\\]fo... | We need solve equation. Need final with boxed. Let's analyze carefully.
Equation: (2x+4)/(x^2+4x-5) = (2-x)/(x-1).
Domain: denominators nonzero. x^2+4x-5 = (x+5)(x-1). x-1 nonzero. So x ≠ 1, -5.
Simplify LHS: (2x+4)=2(x+2). Denom (x+5)(x-1). RHS (2-x)/(x-1) = -(x-2)/(x-1).
Since x≠1, can multiply both sides by (x-1... | We need solve
\[
\frac{2x+4}{x^2+4x-5}=\frac{2-x}{x-1}.
\]
First note the restrictions from the denominators:
\[
x^2+4x-5=(x+5)(x-1),
\]
so
\[
x\neq 1,\qquad x\neq -5.
\]
Factor the left side:
\[
\frac{2x+4}{(x+5)(x-1)}=\frac{2-x}{x-1}.
\]
Since \(x\neq 1\), multiply both sides by \(x-1\):
\[
\frac{2x+4}{x+5}=... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Solve \\[\\frac{2x+4}{x^2+4x-5}=\\frac{2-x}{x-1}\\]fo... | boxed | affine_math | math_algebra | math:0515 | math:0515 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Solve \[\frac{2x+4}{x^2+4x-5}=\frac{2-x}{x-1}\]for $x$.<|im_end|>
<|im_start|>assistant
<think>
| We need solve equation. Need final with boxed. Let's analyze carefully.
Equation: (2x+4)/(x^2+4x-5) = (2-x)/(x-1).
Domain: denominators nonzero. x^2+4x-5 = (x+5)(x-1). x-1 nonzero. So x ≠ 1, -5.
Simplify LHS: (2x+4)=2(x+2). Denom (x+5)(x-1). RHS (2-x)/(x-1) = -(x-2)/(x-1).
Since x≠1, can multiply both sides by (x-1... | 91 | 860 | 951 | 5 |
math_counting_probability.c28e803a.pr_1003196__aadb8a5a:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "When a certain biased coin is flipped five times, the... | We need solve problem. We already called. Need reason step by step plain text, final boxed only. Let's solve.
Problem: biased coin flipped 5 times. P(exactly once heads) not equal 0 and same as P(exactly twice heads). Let i/j in lowest terms be probability heads exactly 3 out of 5. Find i+j.
Let p = P(head), q=1-p. C... | Let \(p\) be the probability of heads and \(q=1-p\) be the probability of tails.
We are told that the probability of exactly one head equals the probability of exactly two heads, and both are nonzero:
\[
5pq^4 = 10p^2q^3
\]
Since \(p,q \neq 0\), divide both sides by \(5pq^3\):
\[
q = 2p
\]
But \(q = 1-p\), so
\[
... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "When a certain biased coin is flipped five times, the... | boxed | affine_math | math_counting_probability | math:0252 | math:0252 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
When a certain biased coin is flipped five times, the probability of getting heads exactly once is ... | We need solve problem. We already called. Need reason step by step plain text, final boxed only. Let's solve.
Problem: biased coin flipped 5 times. P(exactly once heads) not equal 0 and same as P(exactly twice heads). Let i/j in lowest terms be probability heads exactly 3 out of 5. Find i+j.
Let p = P(head), q=1-p. C... | 127 | 636 | 763 | 5 |
math_counting_probability.c28e803a.pr_1003196__aadb8a5a:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "When a certain biased coin is flipped five times, the... | We need solve problem. We already called. Need reason step by step plain text, final boxed only. Let's solve.
Problem: biased coin flipped 5 times. P(exactly once heads) not equal 0 and same as P(exactly twice heads). Let i/j in lowest terms be probability heads exactly 3 out of 5. Find i+j.
Let p = P(head), q=1-p. C... | Let \(p\) be the probability of heads and \(q=1-p\) be the probability of tails.
We are told that the probability of exactly one head equals the probability of exactly two heads, and both are nonzero:
\[
5pq^4 = 10p^2q^3
\]
Since \(p,q \neq 0\), divide both sides by \(5pq^3\):
\[
q = 2p
\]
But \(q = 1-p\), so
\[
... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "When a certain biased coin is flipped five times, the... | boxed | affine_math | math_counting_probability | math:0252 | math:0252 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
When a certain biased coin is flipped five times, the probability of getting heads exactly once is ... | We need solve problem. We already called. Need reason step by step plain text, final boxed only. Let's solve.
Problem: biased coin flipped 5 times. P(exactly once heads) not equal 0 and same as P(exactly twice heads). Let i/j in lowest terms be probability heads exactly 3 out of 5. Find i+j.
Let p = P(head), q=1-p. C... | 127 | 636 | 763 | 5 |
math_number_theory.36271066.pr_6444594__1aa505a6:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Billy and Bobbi each selected a positive integer less... | We need answer math problem. Need solve step by step plain text, final boxed. Let's analyze carefully.
Problem: Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, Bobbi's number is a multiple of 24. What is probability they selected same number? Express common fraction.... | Billy’s possible numbers are the positive multiples of 18 less than 200:
\[
18, 36, \ldots, 198
\]
Since \(198 = 18 \cdot 11\), Billy has 11 possible numbers.
Bobbi’s possible numbers are the positive multiples of 24 less than 200:
\[
24, 48, \ldots, 192
\]
Since \(192 = 24 \cdot 8\), Bobbi has 8 possible numbers.... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Billy and Bobbi each selected a positive integer less... | boxed | affine_math | math_number_theory | math:0146 | math:0146 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18,... | We need answer math problem. Need solve step by step plain text, final boxed. Let's analyze carefully.
Problem: Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, Bobbi's number is a multiple of 24. What is probability they selected same number? Express common fraction.... | 115 | 1,155 | 1,270 | 5 |
math_number_theory.36271066.pr_6444594__1aa505a6:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Billy and Bobbi each selected a positive integer less... | We need answer math problem. Need solve step by step plain text, final boxed. Let's analyze carefully.
Problem: Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, Bobbi's number is a multiple of 24. What is probability they selected same number? Express common fraction.... | Billy’s possible numbers are the positive multiples of 18 less than 200:
\[
18, 36, \ldots, 198
\]
Since \(198 = 18 \cdot 11\), Billy has 11 possible numbers.
Bobbi’s possible numbers are the positive multiples of 24 less than 200:
\[
24, 48, \ldots, 192
\]
Since \(192 = 24 \cdot 8\), Bobbi has 8 possible numbers.... | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Billy and Bobbi each selected a positive integer less... | boxed | affine_math | math_number_theory | math:0146 | math:0146 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18,... | We need answer math problem. Need solve step by step plain text, final boxed. Let's analyze carefully.
Problem: Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, Bobbi's number is a multiple of 24. What is probability they selected same number? Express common fraction.... | 115 | 1,155 | 1,270 | 5 |
math_algebra.cd876a0a.pr_4385691__63169481:0 | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Simplify $\\left( \\frac{1}{2k} \\right)^{-2} \\cdot ... | We need answer simplify expression. Now solve. Expression: (1/(2k))^{-2} * (-k)^3.
Simplify step by step. (1/(2k))^{-2} = (2k)^2 = 4k^2 (assuming k nonzero). (-k)^3 = -k^3. Product = 4k^2 * (-k^3) = -4k^5. Need final boxed. Ensure no extra after. Could mention k != 0 due original denominator. Final \boxed{-4k^5}.
| \[
\left( \frac{1}{2k} \right)^{-2} \cdot (-k)^3
= (2k)^2 \cdot (-k^3)
= 4k^2 \cdot (-k^3)
= -4k^5
\]
\[
\boxed{-4k^5}
\] | [
{
"role": "system",
"content": "Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\\boxed{}`. Emit exactly one `\\boxed{}` block and nothing after it."
},
{
"role": "user",
"content": "Simplify $\\left( \\frac{1}{2k} \\right)^{-2} \\cdot ... | boxed | affine_math | math_algebra | math:0326 | math:0326 | <|im_start|>system
Solve the math problem. Reason step by step in plain text, then end your response with the final answer in `\boxed{}`. Emit exactly one `\boxed{}` block and nothing after it.<|im_end|>
<|im_start|>user
Simplify $\left( \frac{1}{2k} \right)^{-2} \cdot (-k)^3$.<|im_end|>
<|im_start|>assistant
<think>
| We need answer simplify expression. Now solve. Expression: (1/(2k))^{-2} * (-k)^3.
Simplify step by step. (1/(2k))^{-2} = (2k)^2 = 4k^2 (assuming k nonzero). (-k)^3 = -k^3. Product = 4k^2 * (-k^3) = -4k^5. Need final boxed. Ensure no extra after. Could mention k != 0 due original denominator. Final \boxed{-4k^5}.
</thi... | 82 | 198 | 280 | 5 |
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