Write each root using exponents and evaluate. ∛-125

Verified step by step guidance

1. Recognize that the cube root of a number can be expressed using exponents as raising the number to the power of \( \frac{1}{3} \). So, \( \sqrt[3]{-125} = (-125)^{\frac{1}{3}} \).
2. Recall that \( -125 \) can be written as \( -(5^3) \) because \( 5^3 = 125 \). So rewrite the expression as \( (-(5^3))^{\frac{1}{3}} \).
3. Use the property of exponents \( (a^m)^n = a^{m \cdot n} \) to simplify the expression: \( (-(5^3))^{\frac{1}{3}} = - (5^{3 \cdot \frac{1}{3}}) \).
4. Simplify the exponent multiplication \( 3 \cdot \frac{1}{3} = 1 \), so the expression becomes \( -5^1 \).
5. Therefore, the cube root of \( -125 \) simplifies to \( -5 \).

Key Concepts

Cube Roots and Radicals

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is denoted by the radical symbol with an index of 3, such as ∛x. For example, ∛-125 asks for a number which cubed equals -125.

Expressing Roots Using Exponents

Roots can be rewritten as fractional exponents, where the nth root of a number x is expressed as x^(1/n). For the cube root, ∛x is equivalent to x^(1/3). This allows the use of exponent rules to simplify and evaluate roots.

Evaluating Negative Bases with Odd Roots

When taking an odd root (like a cube root) of a negative number, the result is also negative because an odd number of negative factors multiplied together remains negative. Thus, ∛-125 equals -5, since (-5)^3 = -125.