Problem 195 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{5^{2}-3^{2}}{3-5} $$

Step-by-Step Solution

Answer

-8

Step 1: Calculate the Numerator

First, compute the values of the squares in the numerator: \( 5^2 \) and \( 3^2 \). \( 5^2 = 25 \) and \( 3^2 = 9 \). Thus, the numerator is \( 25 - 9 \).

Step 2: Subtract the Squares

Next, subtract the values obtained: \( 25 - 9 = 16 \). This gives the simplified numerator.

Step 3: Calculate the Denominator

Now, compute the denominator: \( 3 - 5 = -2 \).

Step 4: Simplify the Fraction

Finally, combine the simplified numerator and denominator to simplify the fraction: \( \frac{16}{-2} \). By performing the division, we get \( \frac{16}{-2} = -8 \).

Key Concepts

Numerator and Denominator

Let's start by understanding what the numerator and denominator are. The numerator is the number at the top of a fraction, and it represents the part of the whole we have. The denominator is the number at the bottom, showing how many parts the whole is divided into. For the given exercise: \[ \frac{5^{2}-3^{2}}{3-5} \] The numerator is \(5^{2}-3^{2}\), and the denominator is \(3-5\). It's important to compute them separately to simplify the fraction step by step.

Exponentiation

Exponentiation involves raising a number to the power of another. Here, we have \(5^2\) and \(3^2\). When you raise a number to the power of two, you are multiplying that number by itself: \(5^2 = 5 \times 5 = 25\) and \(3^2 = 3 \times 3 = 9\). So, \(25 - 9\) simplifies to \(16\). Thus, our numerator simplifies to 16. This step shows how exponentiation helps simplify expressions.

Fraction Simplification

Now, let's simplify the fraction. After computing the numerator \(25 - 9 = 16\) and the denominator \(3 - 5 = -2\), we combine them to form the fraction \( \frac{16}{-2} \). Fraction simplification involves dividing the numerator by the denominator. Here, \( \frac{16}{-2} \) simplifies directly to -8. It's crucial to perform these operations with care to avoid mistakes.

Negative Numbers

Dealing with negative numbers is essential in simplifications. In our problem, the denominator is negative: \(3 - 5 = -2\). When simplifying \( \frac{16}{-2} \), it's important to recognize that dividing a positive by a negative results in a negative outcome: \( \frac{16}{-2} = -8 \). Understanding this rule helps avoid confusion and ensures accurate results.