Write each root using exponents and evaluate. ∛125

Verified step by step guidance

1. Recognize that the cube root symbol ∛125 means finding a number which, when raised to the power of 3, equals 125.
2. Rewrite the cube root using exponents: the cube root of a number \(a\) can be written as \(a^{\frac{1}{3}}\). So, \(\sqrt[3]{125} = 125^{\frac{1}{3}}\).
3. Express 125 as a power of a smaller base if possible. Since \(125 = 5^3\), rewrite the expression as \(\left(5^3\right)^{\frac{1}{3}}\).
4. Use the power of a power property of exponents: \(\left(a^m\right)^n = a^{m \times n}\). So, \(\left(5^3\right)^{\frac{1}{3}} = 5^{3 \times \frac{1}{3}}\).
5. Simplify the exponent multiplication: \(3 \times \frac{1}{3} = 1\), so the expression becomes \(5^1\), which is simply 5.

Key Concepts

Radicals and Roots

A radical expression involves roots, such as square roots or cube roots, which represent the inverse operation of exponentiation. The cube root (∛) of a number is the value that, when raised to the third power, gives the original number. Understanding radicals helps in rewriting and simplifying root expressions.

Exponential Notation for Roots

Roots can be expressed using fractional exponents, where the nth root of a number is written as that number raised to the power of 1/n. For example, the cube root of 125 can be written as 125^(1/3). This notation allows the use of exponent rules to simplify and evaluate roots.

Evaluating Powers and Roots

Evaluating roots or powers involves calculating the numerical value by applying exponentiation rules. For instance, since 5^3 = 125, the cube root of 125 is 5. Recognizing perfect powers helps in quickly finding the value of roots without a calculator.