Problem 50 - Elementary and Intermediate Algebra

Question

Simplify. $$ \frac{5^{2}-3^{2}}{2 \cdot 6-4} $$

Step-by-Step Solution

Answer

2

Step 1: Calculate the numerator

The numerator is composed of two squared terms: \[5^2 - 3^2\]. Calculate each squared term:\[5^2 = 25\]\[3^2 = 9\]. Subtract the results: \[25 - 9 = 16\].

Step 2: Calculate the denominator

The denominator consists of an arithmetic operation: \[2 \times 6 - 4\]. First, perform the multiplication:\[2 \times 6 = 12\]. Then, subtract 4: \[12 - 4 = 8\].

Step 3: Divide the results

Now, divide the result of the numerator by the result of the denominator: \[\frac{16}{8} = 2\].

Key Concepts

Numerator

In any fraction, the numerator is the top part. It tells you how many parts of the denominator you have. In our example, the numerator is the algebraic expression: \[5^2 - 3^2\].
To simplify it:

• First, calculate each squared term. For example:\[5^2 = 25\] and \[3^2 = 9\].
• Subtract these results to get: \[25 - 9 = 16\].

Denominator

The denominator is the bottom part of a fraction. It represents the total number of equal parts. In the given exercise, the denominator is represented by: \[2 \times 6 - 4\].
To simplify it:

• First, perform the multiplication: \[2 \times 6 = 12\].
• Next, subtract 4 from the result: \[12 - 4 = 8\].

So, the denominator becomes 8.

Arithmetic Operations

In algebra, arithmetic operations include addition, subtraction, multiplication, and division. These operations are essential when simplifying algebraic expressions.

• In our example, we performed addition, subtraction, and multiplication.
• First, we calculated the squares and then subtracted in the numerator: \[25 - 9 = 16\].
• Then, in the denominator, we multiplied and subtracted: \[2 \times 6 - 4 = 8\].

Remember to follow the order of operations (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction, often abbreviated as PEMDAS).

Squared Terms

Squared terms are expressions squared, meaning raised to the power of 2. For instance,

• \[5^2\] means \[5\times5 = 25\].
• Similarly, \[3^2\] means \[3\times3 = 9\].
In our problem, you noticed these squared terms in the numerator. These calculations are fundamental when simplifying such expressions.
Once you calculate the squared terms, simplify them by applying the basic arithmetic operations.