Problem 57 - College Algebra

Question

Evaluate each expression or indicate that the root is not a real number. $$\sqrt[3]{-8}$$

Step-by-Step Solution

Answer

The cube root of -8 is -2.

Step 1: Understanding the Cube Root Operation

The cube root of a number \( n \) is a value that when cubed (multiplied by itself twice), gives the original number \( n \). It's denoted as \( \sqrt[3]{n} \). When \( n \) is negative, its cube root is also negative because the cube of a negative number is negative.

Step 2: Evaluating the Cube Root

Here we need to evaluate \( \sqrt[3]{-8} \). The cube root of -8 is -2 because when -2 is cubed \( (-2 \times -2 \times -2) \) it gives -8.

Key Concepts

Real Numbers

Real numbers include all the numbers you normally encounter in math. They are made up of both rational and irrational numbers.
This means they include integers, fractions, and decimals that either terminate or repeat, as well as numbers that have non-repeating, non-terminating decimals (like pi).
Let’s break this down a bit further:

• Rational Numbers: Numbers that can be expressed as a fraction where both the numerator and the denominator are integers. Examples include 1/2, -3, and 4.
• Irrational Numbers: These cannot be written as simple fractions because their decimal goes on forever without repeating. Examples are √2 and π.

Real numbers can be positive, negative, or zero. This makes them incredibly useful for representing quantities in the real world, such as measurements, money, and time.
When you work with real numbers in algebra, operations like addition, subtraction, multiplication, division, and taking roots are common.

Negative Numbers

Negative numbers are numbers less than zero. You usually see these numbers represented with a minus sign (-).
They are useful in a variety of contexts:

• Temperature: Below zero temperatures are described using negative numbers.
• Finance: If you owe money, your account might show a negative balance.
• Coordinates: Number lines and graphs often use negative numbers to represent points in different directions.

When working with cube roots, negative numbers behave differently than square roots. While the square root of a negative number is not a real number (it's an imaginary number), the cube root of a negative number is real and negative.
This is because multiplying a negative number by itself twice results in a negative product (since two negative multiplications result in a positive product, the third multiplication makes it negative again). In our example, \(-2 \times -2 \times -2 = -8\), showing that \(\sqrt[3]{-8} = -2\).

Algebraic Expressions

Algebraic expressions are combinations of symbols and numbers using operations like addition, subtraction, multiplication, division, and taking roots.
They are like mathematical phrases or sentences without an equal sign (if there's an equal sign, it's an equation).
For example:

• \(2x + 5\): An expression that changes value depending on what \(x\) is.
• \(3a^2 - 4b\): Another expression involving powers and multiple variables.

In the given exercise, \(\sqrt[3]{-8}\) could be part of a larger expression. Understanding how cube roots fit into algebraic expressions helps in solving equations and simplifying complex expressions.
When dealing with roots in algebra, remember:

• Cube Roots: Work with both negative and positive values.
• Operations: Always follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

Algebraic expressions give you the flexibility to explore and solve a wide range of mathematical problems efficiently.