Problem 57 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$\sqrt[3]{-8}$$

Step-by-Step Solution

Answer

-2

Step 1: Recognize the Cube Root

The problem is asking to find the cube root of -8, written as \( \sqrt[3]{-8} \). This means finding the number that, when multiplied by itself three times, gives -8.

Step 2: Evaluate Cube Root

The cube root of -8 is -2 because when you multiply -2 by itself three times \((-2*-2*-2)\), it gives -8.

Key Concepts

Negative Numbers

Dealing with negative numbers can be a bit tricky, but once you understand the basics, it becomes easier. A negative number is simply a number that is less than zero, expressed with a minus sign (-). Here are a couple of key points to consider:

• When you multiply two negative numbers, the result is positive. For example, (-2) * (-2) = 4.
• However, when you multiply a positive number by a negative number, the result is negative, like 2 * (-2) = -4.

These principles are important when evaluating cube roots of negative numbers. Understanding the behavior of negative numbers will help simplify complex algebraic problems.

Cube Root Evaluation

Cube root evaluation involves finding the number which, when multiplied by itself three times (cubed), equals the original number. For example, the cube root of a number is symbolized as \(\sqrt[3]{x}\). Let's see how this applies to negative values:

• In the case of \(\sqrt[3]{-8}\), we want to find a number that multiplies by itself three times to result in -8.
• The suitable number is -2 because (-2) * (-2) * (-2) = -8.
• This confirms that \(\sqrt[3]{-8} = -2\).

Unlike square roots, cube roots can indeed be negative because three negative factors multiply to give a negative result. This is an integral part of calculating expressions involving cube roots.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. Understanding how to manipulate these expressions is crucial for solving many mathematical problems. Here are some basics:

• Expressions can include operations such as addition, subtraction, multiplication, division, and exponentiation.
• When dealing with cube roots like \(\sqrt[3]{-8}\), you'll often break down the expression into simpler terms, as seen when calculating that the cube root of -8 is -2.
• It’s important to apply the rules of negative numbers and exponents to solve algebraic expressions accurately.

This understanding helps not only in simple evaluations but also in more complex sequences of algebra that you will encounter as you progress in mathematics.