Write each root using exponents and evaluate. ∛216

Verified step by step guidance

1. Recognize that the cube root symbol ∛216 means finding a number that, when raised to the power of 3, equals 216.
2. Rewrite the cube root using exponents: the cube root of a number \(a\) can be expressed as \(a^{\frac{1}{3}}\). So, \(\sqrt[3]{216} = 216^{\frac{1}{3}}\).
3. Factor 216 into its prime factors to simplify the expression. For example, find the prime factorization of 216.
4. Express 216 as a product of prime factors raised to powers, then apply the exponent \(\frac{1}{3}\) to each factor using the property \((a^m)^n = a^{m \times n}\).
5. Simplify the expression by multiplying the exponents and then calculate the resulting value to find the cube root.

Key Concepts

Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, etc. The cube root (∛) of a number is the value that, when multiplied by itself three times, equals the original number. Understanding how to interpret and manipulate these roots is essential for solving the problem.

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 216 can be written as 216^(1/3). This notation helps in simplifying and evaluating roots using exponent rules.

Evaluating Powers and Roots

Evaluating roots or fractional exponents involves finding the number that satisfies the power relationship. For ∛216, you find the number which, when cubed, equals 216. Recognizing perfect cubes and performing calculations accurately is key to obtaining the correct value.