Question

Simplify by factoring. $$ \sqrt[3]{-32 a^{6}} $$

   Simplify by factoring.
$$
\sqrt[3]{-32 a^{6}}
$$

Elementary and Intermediate Algebra

Elementary and Intermediate Algebra

Marvin L. Bittinger,… 4th Edition

Chapter 10, Problem 40

Instant Answer

Solved by Expert Cory Kuzinski

05/17/2020

Step 1

In this case, -8 and $a^{6}$ are perfect cubes. We can rewrite the expression as follows: $$ \sqrt[3]{-8 \cdot 4 \cdot a^{6}} $$ Show more…

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Simplify by factoring. $$ \sqrt[3]{-32 a^{6}} $$

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Key Concepts

Cube Roots and Radicals

Cube roots represent a type of radical where you are looking for a number that, when raised to the third power, equals the given expression. They follow similar rules to square roots but are unique in that cube roots of negative numbers are also negative, which makes the operation valid over all real numbers.

Properties of Exponents

The properties of exponents, such as the rules for multiplying and dividing powers and raising a power to another power, are crucial when simplifying expressions that involve variables raised to an exponent. They allow you to break down, combine, or redistribute powers, which is essential when factoring out perfect cubes from an expression.

Factoring Perfect Cubes

Factoring perfect cubes involves identifying parts of the expression that can be rewritten as the cube of a number or a variable expression. This is useful for simplifying cube root expressions because it allows for the extraction of these 'perfect cube' factors from under the radical sign.

Handling Negatives in Cube Roots

Unlike square roots, cube roots can directly handle negative values because the cube of a negative number is negative. This property is essential for simplifying expressions that involve the cube root of a negative number, ensuring that the sign is preserved correctly in the simplification process.

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