Question

finding the cube root of a perfect cube monomial

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

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

Explanation:

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.

Answer:

• (A) \(5x^2\)
• (B) \(5x^4\) (Correct answer)
• (C) \(25x^2\)
• (D) \(25x^4\)