Problem 70 - Intermediate Algebra

Question

Evaluate each expression. See Example \(8 .\) $$ 4^{2}-(-8)^{2} $$

Step-by-Step Solution

Answer

-48

Step 1: Evaluate the first exponent

Calculate \(4^2\). The expression \(4^2\) means 4 multiplied by itself: \(4 \times 4 = 16\). Thus, \(4^2 = 16\).

Step 2: Evaluate the second exponent

Calculate \((-8)^2\). Note that squaring a negative number results in a positive number: \((-8) \times (-8) = 64\). Therefore, \((-8)^2 = 64\).

Step 3: Calculate the expression

Subtract the result of the second exponent from the result of the first exponent: \(16 - 64\). This simplifies to \(-48\).

Key Concepts

Exponents

Exponents, sometimes referred to as powers, represent repeated multiplication of a number by itself. For example, the expression \(4^2\) is equivalent to \(4 \times 4\), which equals 16. The number 4 in \(4^2\) is called the base, and 2 is the exponent. It tells us how many times the base is used in the multiplication.

In general:

• The expression \(a^n\) means the base \(a\), is multiplied by itself \(n\) times.
• An exponent of 2 is known as "squared," and an exponent of 3 is referred to as "cubed."

Exponents provide a concise way to deal with large calculations and make mathematical expressions more manageable. With practice, evaluating exponents becomes intuitive and straightforward. Remember, the key steps are identifying the base and the exponent and then performing the repeated multiplication.

Negative Numbers

Negative numbers are numbers less than zero. They are typically represented with a minus sign (-). Understanding how to work with negative numbers is crucial, especially when they appear in exponents. When a negative number is raised to an even exponent, the result is always positive. For example,

• \((-2)^2 = (-2) \times (-2) = 4\), which is positive.
• This principle applies to any negative number with an even exponent.

On the other hand, if a negative number is raised to an odd exponent, the result is negative again:

• \((-2)^3 = (-2) \times (-2) \times (-2) = -8\).

Always pay attention to parentheses; they indicate that the entire negative number, including the negative sign, should be included in the calculation. This distinction is crucial for accurate calculations involving exponents.

Order of Operations

The order of operations is a fundamental concept in mathematics that dictates the sequence in which calculations should be performed to ensure consistency. Remembering this order can be simplified with the acronym PEMDAS:

• **P**arentheses
• **E**xponents
• **M**ultiplication and **D**ivision (from left to right)
• **A**ddition and **S**ubtraction (from left to right)

In operations such as \(4^2 - (-8)^2\), it's crucial to follow PEMDAS to avoid mistakes:

• Notice how we first calculate the exponents \(4^2\) and \((-8)^2\) separately before proceeding with subtraction.
• The result from the exponents, \(16\) and \(64\), are then subtracted: \(16 - 64\).