Question

simplify the expression.

\\((4q^2r^2)(4q^5r^3)\\)

\\(?q^{\box}r^{\box}\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Simplifying Monomials"
],
"new_concepts": [
"Product Rule of Exponents"
],
"current_concepts": [
"Simplifying Monomials",
"Product Rule of Exponents"
]
}
</pre_analysis>

<reasoning>

Identify the components of the expression

We are given the expression:
\[ (4q^2r^2)(4q^5r^3) \]
To simplify this, we group the coefficients, the \(q\) terms, and the \(r\) terms together.

Group coefficients and like variables

Using the Simplifying Monomials knowledge point
\[
(4 \cdot 4) \cdot (q^2 \cdot q^5) \cdot (r^2 \cdot r^3)
\]

Multiply the coefficients

Using the Simplifying Monomials knowledge point
\[
4 \cdot 4 = 16
\]

Apply the product rule for exponents

To multiply powers with the same base, we add their exponents:
\[
q^2 \cdot q^5 = q^{2+5} = q^7
\]
\[
r^2 \cdot r^3 = r^{2+3} = r^5
\]

Combine the simplified parts

Using the Simplifying Monomials knowledge point
\[
16q^7r^5
\]
</reasoning>

<answer>
Simplify the expression.
\[ (4q^2r^2)(4q^5r^3) \]
<blank>16</blank>\(q\)<blank>7</blank>\(r\)<blank>5</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Simplifying Monomials"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Simplifying Monomials"
],
"new_concepts": [
"Product Rule of Exponents"
],
"current_concepts": [
"Simplifying Monomials",
"Product Rule of Exponents"
]
}
</pre_analysis>

<reasoning>

Identify the components of the expression

We are given the expression:
\[ (4q^2r^2)(4q^5r^3) \]
To simplify this, we group the coefficients, the \(q\) terms, and the \(r\) terms together.

Group coefficients and like variables

Using the Simplifying Monomials knowledge point
\[
(4 \cdot 4) \cdot (q^2 \cdot q^5) \cdot (r^2 \cdot r^3)
\]

Multiply the coefficients

Using the Simplifying Monomials knowledge point
\[
4 \cdot 4 = 16
\]

Apply the product rule for exponents

To multiply powers with the same base, we add their exponents:
\[
q^2 \cdot q^5 = q^{2+5} = q^7
\]
\[
r^2 \cdot r^3 = r^{2+3} = r^5
\]

Combine the simplified parts

Using the Simplifying Monomials knowledge point
\[
16q^7r^5
\]
</reasoning>

<answer>
Simplify the expression.
\[ (4q^2r^2)(4q^5r^3) \]
<blank>16</blank>\(q\)<blank>7</blank>\(r\)<blank>5</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Simplifying Monomials"
]
}
</post_analysis>