Question

write the following expression in simplified radical form.

\\\sqrt5{128x^{12}w^5}\\

assume that all of the variables in the expression represent positive real numbers.

Explanation:

Factor the radicand into perfect fifth powers

Using the Simplifying Complex Radical Expressions knowledge point

$$
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Group the perfect fifth powers together

Using the Simplifying Complex Radical Expressions and Exponent Rules knowledge points

$$ 128x^{12}w^5 = (2^5 \cdot x^{10} \cdot w^5) \cdot (4x^2) = (2x^2w)^5 \cdot (4x^2) $$

Apply the product rule for radicals

Using the Simplifying Complex Radical Expressions knowledge point

$$ \sqrt[5]{128x^{12}w^5} = \sqrt[5]{(2x^2w)^5} \cdot \sqrt[5]{4x^2} $$

Simplify the perfect fifth root

Using the Simplifying Complex Radical Expressions knowledge point

$$ \sqrt[5]{(2x^2w)^5} \cdot \sqrt[5]{4x^2} = 2x^2w\sqrt[5]{4x^2} $$

Answer:

Write the following expression in simplified radical form.

\(\sqrt[5]{128x^{12}w^5}\)

Assume that all of the variables in the expression represent positive real numbers.

<blank>\(2x^2w\sqrt[5]{4x^2}\)</blank>