Problem 112 - Intermediate Algebra

Question

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ (-8)^{1 / 3} $$

Step-by-Step Solution

Answer

The correct letter for \((-8)^{1/3}\) is B, as it equals -2.

Step 1: Understand the Expression

The expression given is \[ (-8)^{1 / 3} \] which means "the cube root of -8." We are asked to determine if this expression evaluates to 2, -2, or is not a real number.

Step 2: Define the Cube Root

The cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \). In our case, we are looking for a number \( y \) where \( y^3 = -8 \).

Step 3: Solve for the Cube Root

Calculate \[ (-2)^3 = -2 \times -2 \times -2 = -8 \] Thus, the cube root of \(-8\) is \(-2\).

Step 4: Choose the Correct Letter

Based on the options given, -2 corresponds to B. Thus, \[ (-8)^{1/3} = -2 \] is represented by option B.

Key Concepts

Understanding Exponents

Exponents are a mathematical concept that show how many times a number, called the base, is multiplied by itself. They are written as a superscript number to the right of the base. For example, in the expression \(3^4\), the number 3 is the base and 4 is the exponent, indicating that 3 is multiplied by itself four times (\(3 \times 3 \times 3 \times 3 = 81\)).

Exponents have several rules, such as:

• The product of powers rule: \(a^m \times a^n = a^{m+n}\).
• The power of a power rule: \((a^m)^n = a^{m\times n}\).
• The power of a product rule: \((ab)^n = a^n \times b^n\).

These rules help simplify and solve expressions involving powers.

When dealing with fractional exponents, like \(a^{1/3}\), this represents a root. Specifically, \(a^{1/3}\) is the cube root of \(a\). Cube roots are unique because they allow us to find a number \(y\) such that \(y^3 = a\). Being comfortable with these concepts can greatly aid in solving equations involving exponents.

Grasping Real Numbers

Real numbers include all the numbers that can be found on the number line. This collection encompasses different types of numbers such as whole numbers, fractions, decimals, and irrational numbers (like \(\sqrt{2}\) and \(\pi\)). In essence, any number you can think of likely falls into the category of real numbers.

Real numbers have specific properties:

• They are dense, meaning between any two real numbers, there is another real number.
• They can be positive, negative, or zero.
• They follow standard arithmetic operations including addition, subtraction, multiplication, and division (except by zero).

Real numbers are crucial in understanding various mathematical concepts and are used extensively in daily life and complex scientific computations alike.

Navigating Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a minus sign. These numbers have noteworthy features and behaviors, especially when it comes to mathematical operations.

When multiplying or dividing negative numbers, keep these rules in mind:

• The product or quotient of two negative numbers is positive: \((-a) \times (-b) = ab\).
• The product or quotient of a positive and a negative number is negative: \(a \times (-b) = -ab\).

Additionally, negative numbers are vital in expressing temperatures below zero, debts, depths below sea level, and more.

Understanding negative numbers also involves mastering root operations, such as the cube root. Unlike square roots, the cube root of a negative number is also negative. For example, the cube root of -8 is -2, because \((-2)^3 = -2 \times -2 \times -2 = -8\). This property is particularly useful in solving equations where negative values play a critical role.