Question

what is \\(64x^6 + 27\\) written as a sum of cubes?

\\((4x)^3 + 3^3\\)
\\((4x^2)^3 + 3^3\\)
\\((4x^2)^3 + 9^3\\)
\\((4x^3)^3 + 3^3\\)

Explanation:

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<reasoning>

Rewrite the first term as a perfect cube

Using the Cube Root of Monomials and Perfect Cubes knowledge points
\[

$$\begin{aligned} &64x^6 = (4)^3 \cdot (x^2)^3 = (4x^2)^3 \end{aligned}$$

\]

Rewrite the second term as a perfect cube

Using the Perfect Cubes knowledge point
\[

$$\begin{aligned} &27 = 3^3 \end{aligned}$$

\]

Express the polynomial as a sum of cubes

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} &64x^6 + 27 = (4x^2)^3 + 3^3 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((4x)^3 + 3^3\)</mcq-option>
<mcq-correct>(B) \((4x^2)^3 + 3^3\)</mcq-correct>
<mcq-option>(C) \((4x^2)^3 + 9^3\)</mcq-option>
<mcq-option>(D) \((4x^3)^3 + 3^3\)</mcq-option>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Rewrite the first term as a perfect cube

Using the Cube Root of Monomials and Perfect Cubes knowledge points
\[

$$\begin{aligned} &64x^6 = (4)^3 \cdot (x^2)^3 = (4x^2)^3 \end{aligned}$$

\]

Rewrite the second term as a perfect cube

Using the Perfect Cubes knowledge point
\[

$$\begin{aligned} &27 = 3^3 \end{aligned}$$

\]

Express the polynomial as a sum of cubes

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} &64x^6 + 27 = (4x^2)^3 + 3^3 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((4x)^3 + 3^3\)</mcq-option>
<mcq-correct>(B) \((4x^2)^3 + 3^3\)</mcq-correct>
<mcq-option>(C) \((4x^2)^3 + 9^3\)</mcq-option>
<mcq-option>(D) \((4x^3)^3 + 3^3\)</mcq-option>
</answer>

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"Sum of Cubes"
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