Problem 28 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 3^{2}-2^{4} $$

Step-by-Step Solution

Answer

The simplified expression is -7.

Step 1: Evaluate the Exponentials

We start by calculating the values of the exponentials in the expression. The expression is \(3^2 - 2^4\). First, calculate \(3^2 = 3 \times 3 = 9\) and \(2^4 = 2 \times 2 \times 2 \times 2 = 16\).

Step 2: Perform the Subtraction

Now that we have evaluated the exponentials, we substitute them back into the expression. This gives us \(9 - 16\). Perform the subtraction: \(9 - 16 = -7\).

Key Concepts

Exponentiation

When dealing with exponentiation, it's important to understand what the term means. Exponentiation refers to the operation of raising a base number to the power of an exponent. It is a way to express repeated multiplication concisely. For example, in the expression \(3^2\), the number 3 is the base, and 2 is the exponent. This means you multiply the base by itself as many times as indicated by the exponent. So, \(3^2\) translates to \(3 \times 3\), resulting in 9.

To simplify expressions with exponents, follow these steps:

• Identify the base and the exponent.
• Multiply the base by itself the number of times specified by the exponent.
• Write down the resulting value to replace the exponential term in your original expression.

Understanding exponentiation as repeated multiplication is crucial, especially when simplifying more complex expressions.

Integer Operations

Integer operations refer to arithmetic operations involving whole numbers which can include both positive and negative values. In the context of simplifying expressions, integer operations will often involve addition, subtraction, multiplication, and sometimes division of integers.

For the task at hand, the operations include subtraction, which is the process of finding the difference between two numbers. After calculating the exponentials in the given expression \(3^2 - 2^4\), we obtain \(9 - 16\). Here, the integer operation involves subtracting 16 from 9, which requires recognizing that you are working with different magnitudes and potentially involving negative results.

To carry out integer operations successfully:

• Understand the sign of each number in your problem.
• Apply the basic arithmetic rules for addition and subtraction.
• Keep in mind that subtracting a larger number from a smaller number will yield a negative result.

Integer operations form the basis for handling more complex expressions and equations, and are a fundamental mathematical skill.

Negative Numbers

Negative numbers represent values less than zero and are often found on the left side of a number line. They are crucial when simplifying expressions that result in values below zero, as seen with the subtraction \(9 - 16\) which equals \(-7\).

Dealing with negative numbers requires understanding how they interact with other numbers in different arithmetic contexts:

• Subtracting a positive number from a smaller positive number yields a negative result.
• Adding a negative number to a positive number essentially involves subtracting the absolute values.
• Negative numbers multiplied by positive numbers will result in negative products.

Grasping the concept of negative numbers is essential not only for academic exercises but also for real-world applications such as calculating debts or temperature changes. They allow you to express and work with values that are less than zero logically and consistently.