Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ³√8

Verified step by step guidance

1. Identify the expression to evaluate: the cube root of 8, written as \(\sqrt[3]{8}\).
2. Recall that the cube root of a number \(a\) is a number \(b\) such that \(b^3 = a\).
3. Determine which number, when raised to the power of 3, equals 8. In other words, solve \(b^3 = 8\).
4. Recognize that \(2^3 = 2 \times 2 \times 2 = 8\), so \(b = 2\) satisfies the equation.
5. Conclude that the cube root of 8 is a real number and is equal to 2.

Key Concepts

Cube Roots

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8. Cube roots can be positive, negative, or zero.

Real Numbers and Roots

Real numbers include all rational and irrational numbers. When evaluating roots, it is important to determine if the root is real or not. Cube roots of any real number always exist and are real, unlike even roots which may not be real for negative numbers.

Evaluating Radical Expressions

Evaluating radical expressions involves simplifying the root to its simplest form. This requires understanding how to rewrite numbers as powers and applying root properties. For cube roots, this means finding the number that raised to the third power equals the given value.