Problem 83 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \(\frac{3^{2}-4^{2}}{7(-8+9)}\)

Step-by-Step Solution

Answer

The result is simplified to -25.

Step 1: Simplify the exponents

Calculate the exponents in the numerator first. Here, we have two exponents: – – – Begin by calculating the squares: – – – –

Step 2: Subtract the results in the numerator

Now subtract the results obtained from the exponentiation in the numerator: – – – –

Step 3: Simplify the expression in the denominator

Calculate the expression inside the parentheses: – – Inside the parentheses, perform the calculation: – –

Step 4: Multiply the result in the denominator

Multiply the result obtained from the parentheses with 7, as indicated by the expression: – –

Step 5: Divide the simplified numerator by the simplified denominator

Finally, divide the simplified numerator by the simplified denominator: – –

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation involving raising a number, called the base, to the power of an exponent. For example, in the expression \(3^2\), 3 is the base and 2 is the exponent. It tells us to multiply the base by itself a certain number of times indicated by the exponent. So, \(3^2 = 3 \times 3 = 9\).

Similarly, for \(4^2\), the base is 4 and the exponent is 2, and it means: \( 4 \times 4 = 16 \). Mastering this operation is crucial when dealing with polynomial expressions and various other mathematical problems.

Numerator and Denominator

In any fraction, we have two main components: the numerator and the denominator. They are separated by a division line. The numerator is the top part, and the denominator is the bottom part. For instance, in the fraction \( \frac{3^2 - 4^2}{7(-8 + 9)} \), \(3^2 - 4^2\) is the numerator and \(7(-8 + 9)\) is the denominator.

The numerator indicates how many parts we are considering, while the denominator shows the total number of equal parts in the whole. Correctly identifying and simplifying these parts is essential in resolving fractions.

In this exercise, we first simplified the exponents in the numerator and simplified the expression in the parentheses in the denominator, to facilitate the overall calculation.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining the same value. This often means performing basic arithmetic operations like addition, subtraction, multiplication, and division. Here's a step-by-step example based on the original exercise:

• First, calculate the exponents: \(3^2 = 9\) and \(4^2 = 16\).
• Subtract these results in the numerator: \(9 - 16 = -7\).
• Simplify the expression in the denominator: calculate inside the parentheses first: \(-8 + 9 = 1\).
• Then multiply by 7: \(7 \times 1 = 7\).
• Finally, divide the simplified numerator by the simplified denominator: \(\frac{-7}{7} = -1\)

Always follow these steps to avoid mistakes and ensure your final result is correct. Practicing this process will make you more confident and efficient in dealing with polynomial operations.