Question

use the product property of roots to choose the expression equivalent to \\(\sqrt3{5x} \cdot \sqrt3{25x^2}\\)

\\(\sqrt3{30x}\\)

\\(\sqrt3{125x^3}\\)

\\(\sqrt3{30x^2}\\)

\\(\sqrt6{125x^3}\\)

Explanation:

Apply the product property of radicals

Using the product property of roots, \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\):

$$ \sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{5x \cdot 25x^2} $$

Simplify the radicand

Multiply the coefficients and the variables inside the cube root:

$$ 5x \cdot 25x^2 = (5 \cdot 25) \cdot (x \cdot x^2) = 125x^3 $$

Write the final radical expression

Substitute the simplified radicand back into the cube root:

$$ \sqrt[3]{125x^3} $$

Answer:

• (A) \(\sqrt[3]{30x}\)
• (B) \(\sqrt[3]{125x^3}\) (Correct answer)
• (C) \(\sqrt[3]{30x^2}\)
• (D) \(\sqrt[6]{125x^3}\)