In Exercises 77–90, simplify each expression. Include absolute value bars where necessary. ____ ³√(−5)³

Verified step by step guidance

1. Identify the expression: $\sqrt[3]{(-5{)}^3}$.
2. Recognize that the cube root and the cube are inverse operations.
3. Apply the property: $\sqrt[3]{a^3}=a$ for any real number $a$.
4. Since $a=-5$, substitute into the property: $\sqrt[3]{(-5{)}^3}=-5$.
5. Conclude that the expression simplifies to $-5$.

Key Concepts

Cube Root

The cube root of a number 'x' is a value 'y' such that y³ = x. It is denoted as ³√x. Unlike square roots, cube roots can be taken of negative numbers, resulting in a negative output. For example, ³√(-5) means finding a number that, when multiplied by itself three times, equals -5.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base is multiplied by itself. In the expression (−5)³, -5 is the base and 3 is the exponent, meaning -5 is multiplied by itself three times, resulting in -125.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, e.g., |x|. For negative numbers, the absolute value converts them to positive. In the context of cube roots, while the cube root of a negative number is negative, the absolute value would represent its positive counterpart.