Question

simplify the expression.

\\v^{-3} 8^0 x^9\\

\\\frac{?^{ }}{ ^{ }}\\

Explanation:

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

<reasoning>

Identify the given expression

We are given the algebraic expression:
\[
v^{-3} 8^{0} x^{9}
\]
Our goal is to simplify this expression so that it contains only positive exponents.

Apply the zero exponent rule

The zero exponent rule states that any non-zero base raised to the power of \(0\) is equal to \(1\):
\[
a^0 = 1 \quad (a
eq 0)
\]
Applying this rule to \(8^0\):
\[
8^0 = 1
\]
Substituting this back into the expression gives:
\[
v^{-3} \cdot 1 \cdot x^{9} = v^{-3} x^{9}
\]

Apply the negative exponent rule

The negative exponent rule states that a base raised to a negative power is equal to its reciprocal raised to the positive power:
\[
b^{-n} = \frac{1}{b^n} \quad (b
eq 0)
\]
Applying this rule to \(v^{-3}\):
\[
v^{-3} = \frac{1}{v^3}
\]

Combine the terms into a single fraction

Now, we multiply the simplified parts together:
\[
\frac{1}{v^3} \cdot x^9 = \frac{x^9}{v^3}
\]
This expression is fully simplified and contains only positive exponents.
</reasoning>

<answer>
Simplify the expression.

\(v^{-3}8^{0}x^{9} =\) <blank>\(\frac{x^9}{v^3}\)</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"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Zero Exponent Rule",
"Negative Exponent Rule",
"Simplifying Monomials"
],
"current_concepts": [
"Zero Exponent Rule",
"Negative Exponent Rule",
"Simplifying Monomials"
]
}
</pre_analysis>

<reasoning>

Identify the given expression

We are given the algebraic expression:
\[
v^{-3} 8^{0} x^{9}
\]
Our goal is to simplify this expression so that it contains only positive exponents.

Apply the zero exponent rule

The zero exponent rule states that any non-zero base raised to the power of \(0\) is equal to \(1\):
\[
a^0 = 1 \quad (a
eq 0)
\]
Applying this rule to \(8^0\):
\[
8^0 = 1
\]
Substituting this back into the expression gives:
\[
v^{-3} \cdot 1 \cdot x^{9} = v^{-3} x^{9}
\]

Apply the negative exponent rule

The negative exponent rule states that a base raised to a negative power is equal to its reciprocal raised to the positive power:
\[
b^{-n} = \frac{1}{b^n} \quad (b
eq 0)
\]
Applying this rule to \(v^{-3}\):
\[
v^{-3} = \frac{1}{v^3}
\]

Combine the terms into a single fraction

Now, we multiply the simplified parts together:
\[
\frac{1}{v^3} \cdot x^9 = \frac{x^9}{v^3}
\]
This expression is fully simplified and contains only positive exponents.
</reasoning>

<answer>
Simplify the expression.

\(v^{-3}8^{0}x^{9} =\) <blank>\(\frac{x^9}{v^3}\)</blank>
</answer>

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