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 as a cube root: \( \sqrt[3]{8} \).
2. Recall that the cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \).
3. Consider the number 8 and think of a number \( y \) that when multiplied by itself three times equals 8.
4. Recognize that \( 2^3 = 2 \times 2 \times 2 = 8 \).
5. Conclude that the cube root of 8 is 2, since \( 2^3 = 8 \).

Key Concepts

Cube Root

The 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, since 2 × 2 × 2 = 8. This concept is essential for evaluating expressions involving cube roots, as it allows us to determine the real number that satisfies the equation.

Real Numbers

Real numbers include all the numbers on the number line, encompassing both rational numbers (like integers and fractions) and irrational numbers (like √2 and π). Understanding whether a root is a real number is crucial, as some roots, particularly even roots of negative numbers, do not yield real results. In this case, the cube root of 8 is a real number.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are a fundamental part of algebra. They can be simplified or evaluated to find their numerical value. Recognizing how to manipulate and evaluate these expressions is key to solving problems that involve roots, such as the cube root of 8.