Question

simplify the expression.

\\(3x^{12}z^4)^3\\

\\?x^{\box}z^{\box}\\

Explanation:

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<reasoning>

Apply the power of a product rule

Using the Simplifying Monomials knowledge point
\[
(3x^{12}z^4)^3 = 3^3 \cdot (x^{12})^3 \cdot (z^4)^3
\]

Simplify the coefficient

Evaluate the numerical base raised to the power:
\[
3^3 = 3 \cdot 3 \cdot 3 = 27
\]

Apply the power of a power rule

Multiply the exponents for each variable:
\[

$$\begin{aligned} (x^{12})^3 &= x^{12 \cdot 3} = x^{36} \\ (z^4)^3 &= z^{4 \cdot 3} = z^{12} \end{aligned}$$

\]

Combine the simplified terms

Write the final simplified monomial expression:
\[
27x^{36}z^{12}
\]
</reasoning>

<answer>
Simplify the expression.

\[(3x^{12}z^4)^3\]

<blank>\(27\)</blank>\(x\)<blank>\(36\)</blank>\(z\)<blank>\(12\)</blank>
</answer>

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"Mathematics",
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Answer:

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<reasoning>

Apply the power of a product rule

Using the Simplifying Monomials knowledge point
\[
(3x^{12}z^4)^3 = 3^3 \cdot (x^{12})^3 \cdot (z^4)^3
\]

Simplify the coefficient

Evaluate the numerical base raised to the power:
\[
3^3 = 3 \cdot 3 \cdot 3 = 27
\]

Apply the power of a power rule

Multiply the exponents for each variable:
\[

$$\begin{aligned} (x^{12})^3 &= x^{12 \cdot 3} = x^{36} \\ (z^4)^3 &= z^{4 \cdot 3} = z^{12} \end{aligned}$$

\]

Combine the simplified terms

Write the final simplified monomial expression:
\[
27x^{36}z^{12}
\]
</reasoning>

<answer>
Simplify the expression.

\[(3x^{12}z^4)^3\]

<blank>\(27\)</blank>\(x\)<blank>\(36\)</blank>\(z\)<blank>\(12\)</blank>
</answer>

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"Mathematics",
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]
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