Question

finding the cube root of a perfect cube monomial

what is \\(\sqrt3{125x^{12}}\\)?

\\(5x^2\\)
\\(5x^4\\)
\\(25x^2\\)
\\(25x^4\\)

Explanation:

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<reasoning>

Identify the components of the radical expression

We are asked to find the cube root of the monomial expression:
\[
\sqrt[3]{125x^{12}}
\]
This expression consists of a numerical coefficient \(125\) and a variable part \(x^{12}\).

Apply the product property of radicals

Using the product property of radicals, we can separate the cube root of the product into the product of the cube roots:
\[
\sqrt[3]{125x^{12}} = \sqrt[3]{125} \cdot \sqrt[3]{x^{12}}
\]

Simplify the numerical coefficient

Using the Perfect Cubes knowledge point, we recognize that \(125\) is a perfect cube:
\[
125 = 5^3
\]
Therefore, the cube root of \(125\) is:
\[
\sqrt[3]{125} = 5
\]

Simplify the variable part

To find the cube root of a variable raised to a power, we divide the exponent by the index of the radical (which is \(3\)):
\[
\sqrt[3]{x^{12}} = x^{\frac{12}{3}} = x^4
\]

Combine the simplified parts

Multiplying the simplified numerical coefficient and variable part together gives:
\[
5 \cdot x^4 = 5x^4
\]
This matches the second option.
</reasoning>

<answer>
<mcq-option>(A) \(5x^2\)</mcq-option>
<mcq-correct>(B) \(5x^4\)</mcq-correct>
<mcq-option>(C) \(25x^2\)</mcq-option>
<mcq-option>(D) \(25x^4\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the components of the radical expression

We are asked to find the cube root of the monomial expression:
\[
\sqrt[3]{125x^{12}}
\]
This expression consists of a numerical coefficient \(125\) and a variable part \(x^{12}\).

Apply the product property of radicals

Using the product property of radicals, we can separate the cube root of the product into the product of the cube roots:
\[
\sqrt[3]{125x^{12}} = \sqrt[3]{125} \cdot \sqrt[3]{x^{12}}
\]

Simplify the numerical coefficient

Using the Perfect Cubes knowledge point, we recognize that \(125\) is a perfect cube:
\[
125 = 5^3
\]
Therefore, the cube root of \(125\) is:
\[
\sqrt[3]{125} = 5
\]

Simplify the variable part

To find the cube root of a variable raised to a power, we divide the exponent by the index of the radical (which is \(3\)):
\[
\sqrt[3]{x^{12}} = x^{\frac{12}{3}} = x^4
\]

Combine the simplified parts

Multiplying the simplified numerical coefficient and variable part together gives:
\[
5 \cdot x^4 = 5x^4
\]
This matches the second option.
</reasoning>

<answer>
<mcq-option>(A) \(5x^2\)</mcq-option>
<mcq-correct>(B) \(5x^4\)</mcq-correct>
<mcq-option>(C) \(25x^2\)</mcq-option>
<mcq-option>(D) \(25x^4\)</mcq-option>
</answer>

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