Problem 51 - College Algebra

Question

Evaluate each expression in Exercises \(49-60\), or indicate that the root is not a real number. $$\sqrt[3]{-8}$$

Step-by-Step Solution

Answer

-2

Step 1: Understanding Cube Root

The cube root of a number \( x \) is a value that, when cubed (or multiplied by itself twice), gives \( x \). In this case, we are looking for a number that, when cubed, gives -8.

Step 2: Evaluating the cube root

The cube root of -8 is -2 because when -2 is cubed (i.e., \(-2 \times -2 \times -2\)), the result is -8.

Key Concepts

Evaluating Expressions

When we talk about evaluating expressions, we refer to the process of finding the value of a mathematical phrase. This usually involves performing operations like addition, subtraction, multiplication, division, and also dealing with exponents and roots.

For instance, to evaluate the expression \(\sqrt[3]{-8}\), we need to determine which number, when multiplied by itself three times, yields -8. It's crucial to understand that the cube root operation is the inverse of cubing a number. Thus, by understanding the cube root as 'which number cubed gives me my original number?', evaluating expressions becomes a matter of reversing the operations applied to a known result.

It's similar to thinking backward: if cubing a number gives us -8, then the cube root of -8 must be that number. This conceptual approach helps students to link the idea of roots to more intuitive operations like multiplication.

Real Numbers

The concept of real numbers is foundational in mathematics, encompassing both rational numbers (such as fractions and integers) and irrational numbers (which cannot be expressed as fractions). The set of real numbers includes all the numbers we typically use for counting, measuring, and ordering.

Why is this important to our problem? When evaluating the cube root of -8, we must ensure the result is a real number. While the square root of a negative number doesn't yield a real number (and instead introduces us to complex numbers with 'i' for the imaginary unit), cube roots are different. Since raising a negative number to an odd power, like three, keeps the sign negative, the cube root of a negative number still falls within the real numbers. In our exercise, the cube root of -8 is indeed a real number: -2.

Exponentiation

The term exponentiation might sound complex, but it's just the mathematical operation involving the raising of a number, called the base, to the power of an exponent. For example, in the expression \(3^2\), 3 is the base and 2 is the exponent, indicating that you multiply 3 by itself once (since 3 to the power of 1 is simply 3).

In the context of cube roots, we're essentially doing the inverse of exponentiation. While exponentiation would take a base and raise it to the third power (as in \(x^3\)), finding a cube root (\(\sqrt[3]{x}\)) asks us to think about what original number would result in \(x\) after being raised to the third power.

To illustrate with our exercise, \(\sqrt[3]{-8}\) asks us to consider what number, when raised to the power of three (\(x^3\)), would give us -8. The answer, as in our exercise solution, is -2 because \( (-2)^3 = -2 \times -2 \times -2 = -8 \). This relationship between exponentiation and roots is essential for understanding how to evaluate expressions involving them.