In Exercises 77–90, simplify each expression. Include absolute value bars where necessary. __ ³√x³

Verified step by step guidance

1. Recognize that the expression $\sqrt[3]{x^3}$ is asking for the cube root of $x^3$.
2. Recall the property of radicals and exponents: $\sqrt[n]{a^n}=a$ if $n$ is odd.
3. Since the cube root ($n=3$) is an odd root, the expression simplifies directly to $x$.
4. Understand that absolute value bars are not necessary here because the cube root of a number can be negative, unlike even roots.
5. Conclude that the simplified form of $\sqrt[3]{x^3}$ is simply $x$.

Key Concepts

Cube Root

The cube root of a number x, denoted as ³√x, is a value that, when multiplied by itself three times, gives the original number x. For example, ³√8 = 2 because 2 × 2 × 2 = 8. Understanding cube roots is essential for simplifying expressions involving powers and roots.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. When simplifying expressions with roots, especially with variables, it is important to consider absolute values to ensure the result is non-negative, as cube roots can yield both positive and negative results.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms, factoring, and applying properties of exponents and roots. In the context of the given expression ³√x³, recognizing that the cube root and the exponent can cancel each other out is key to achieving a simplified result.