Write each root using exponents and evaluate. ∛-125

Verified step by step guidance

1. Identify the expression: ∛(-125) is asking for the cube root of -125.
2. Recall that the cube root of a number is the same as raising that number to the power of 1/3.
3. Express the cube root using exponents: (-125)^{1/3}.
4. Recognize that -125 can be rewritten as (-5)^3, since (-5) * (-5) * (-5) = -125.
5. Apply the exponent rule: ((-5)^3)^{1/3} = (-5)^{3 * (1/3)} = (-5)^1.

Key Concepts

Roots and Exponents

Roots are the inverse operations of exponents. The nth root of a number 'a' is a value 'b' such that b^n = a. For example, the cube root of -125 can be expressed as -125^(1/3), which indicates that we are looking for a number that, when raised to the power of 3, equals -125.

Negative Numbers and Odd Roots

When dealing with odd roots, such as cube roots, negative numbers can yield real results. Specifically, the cube root of a negative number is also negative. This is in contrast to even roots, where the root of a negative number is not a real number. Thus, ∛-125 will yield a real number.

Evaluation of Roots

To evaluate a root, you determine the number that satisfies the root equation. For ∛-125, you need to find a number that, when multiplied by itself three times, equals -125. In this case, the evaluation leads to -5, since (-5) × (-5) × (-5) = -125.