input stringlengths 112 4k | output stringlengths 36 60.3k | gts stringlengths 1 270 | score float64 0 1 | acc float64 0 1 | step int64 1 63 | uid stringlengths 36 36 | prompt_token_ids listlengths 33 1.02k | response_token_ids listlengths 18 4.1k | rollout_log_probs listlengths 18 4.1k |
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Sure, let's solve each equation step by step.
### Problem (1): \(25x^2 - 9 = 7\)
**Step 1: Isolate the term with \(x\).**
Start by adding 9 to both sides of the equation to isolate the term with \(x^2\):
\[
25x^2 - 9 + 9 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \(x^2\).**
Divide both sides by 25 to solve fo... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### Problem (1): \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
\[
25x^2 - 9 = 7
\]
Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 \)**
Divide both sides by 25:
\[
x^2 = \frac{16}{25}
\]
**S... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve the equation \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
\[
25x^2 - 9 = 7
\]
Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 \)**
Divide both sides by 25:
\[
x^2 = \frac{16}{2... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve the equation \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to isolate the term \( 25x^2 \). To do this, add 9 to both sides of the equation:
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplifying both sides:
\[
25x^2 = 16
\]
**Step 2: Solve for... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve the problems step by step.
### Problem (1): \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to isolate the term \( 25x^2 \). To do this, we add 9 to both sides of the equation.
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplifying both sides:
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 ... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each problem step by step.
### Problem (1): \(25x^2 - 9 = 7\)
**Step 1: Isolate the term with \(x\).**
\[
25x^2 - 9 = 7
\]
Add 9 to both sides to isolate the term with \(x\):
\[
25x^2 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \(x^2\).**
Divide both sides by 25:
\[
x^2 = \frac{16}{25}
\]
**Step 3: ... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Sure, let's solve each equation step by step.
### Problem (1): \(25x^2 - 9 = 7\)
#### Step 1: Isolate the term with \(x\)
First, we need to isolate the term \(25x^2\). To do this, we add 9 to both sides of the equation:
\[25x^2 - 9 + 9 = 7 + 9\]
\[25x^2 = 16\]
#### Step 2: Solve for \(x^2\)
Next, we divide both side... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Sure, let's solve each equation step by step.
### Problem (1): \( 25x^2 - 9 = 7 \)
#### Step 1: Isolate the term with \( x \)
First, we need to isolate the term \( 25x^2 \). To do this, we add 9 to both sides of the equation:
\[ 25x^2 - 9 + 9 = 7 + 9 \]
\[ 25x^2 = 16 \]
#### Step 2: Solve for \( x^2 \)
Next, we divi... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve \( 25x^2 - 9 = 7 \)
**Step 1:** Start with the given equation:
\[
25x^2 - 9 = 7
\]
**Step 2:** Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 - 9 + 9 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 3:** Divide both sides by 25 to solve for \( x^2 \):
\[
\frac... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve the equations step by step.
---
### **Problem (1):**
**Find the value of \( x \):**
\( 25x^2 - 9 = 7 \)
---
#### Step 1: Bring the constant term to the left side
Add \( 9 \) to both sides of the equation to isolate the term with \( x \):
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplify:
\[
25x^2 = 16
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### Problem 1: \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to get rid of the constant term on the left side of the equation. To do this, add 9 to both sides of the equation:
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplify both sides:
\[
25x^2 = 16
\]
*... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve the equation \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
\[
25x^2 - 9 = 7
\]
Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 \)**
Divide both sides by 25:
\[
x^2 = \frac{16}{2... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each problem step by step.
### Problem (1): Solve for \( x \) in the equation \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to isolate the term containing \( x \). Start by adding 9 to both sides of the equation:
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplifying both sides:
\[
25x^... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### Problem (1): \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to isolate the term \( 25x^2 \). To do this, add 9 to both sides of the equation.
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplifying both sides:
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 \)... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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872,
198,
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87,
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve the equation \( 25x^2 - 9 = 7 \)
**Step 1:** Start with the given equation:
\[
25x^2 - 9 = 7
\]
**Step 2:** Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 - 9 + 9 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 3:** Divide both sides by 25 to solve for \( x^2... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
151644,
872,
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400,
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400,
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
---
### **Problem (1):**
Find the value of \( x \) such that
\[ 25x^2 - 9 = 7 \]
---
#### Step 1: Isolate the term with \( x \)
First, add 9 to both sides of the equation to isolate the term containing \( x \).
\[
25x^2 - 9 + 9 = 7 + 9
\]
\[
25x^2 = 16
\]
#### Step 2... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
151644,
872,
198,
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279,
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16,
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20,
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) given the conditions \(\sin \alpha + \sin \beta + \sin \gamma = 0\) and \(\cos \alpha + \cos \beta + \cos \gamma = 0\).
Let's proceed step by step.
### Step 1: Understand the Given Conditions
We are given:
1. \(\sin \a... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
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1124,
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-0.45... |
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