Problem 103 - Algebra 1

Question

EXPRESSIONS WITH FRACTION BARS Evaluate the expression. $$ \frac{6^{2}-12}{3^{2}+15} $$

Step-by-Step Solution

Answer

The value of the given expression is 1

Step 1: Evaluate the Numerator

The numerator involves an exponentiation and a subtraction operation. Start by calculating the exponentiation part which is \(6^2\), then subtract 12 from the result. So, \(6^2 - 12 = 36 - 12 = 24\). Thus, the numerator simplifies to 24.

Step 2: Evaluate the Denominator

The denominator includes an exponentiation and an addition operation. Evaluate \(3^2\) first and then add 15 to the result. So, \(3^2 + 15 = 9 + 15 = 24\). Hence, the denominator simplifies to 24.

Step 3: Simplify the fraction

Now that both the numerator and the denominator have been simplified to 24, the whole fraction simplifies to \(24 / 24 = 1\).

Key Concepts

Numerator and Denominator

In a fraction, we have two important parts: the numerator and the denominator. The numerator is the number on top of the fraction bar, and the denominator is the number on the bottom.
For example, in the fraction \( \frac{3}{4} \), the numerator is 3, and the denominator is 4.

These two components tell us:

• The numerator represents how many parts we have.
• The denominator shows the total number of equal parts something is divided into.

When evaluating fractions with expressions, always resolve each side separately before moving on to simplify.
As seen in the given exercise, the numerator is evaluated by calculating \( 6^2 - 12 \), whereas the denominator is found by calculating \( 3^2 + 15 \). Each part is dealt with in steps to solve the expression effectively.

Exponentiation

Exponentiation is a mathematical operation involving numbers called bases and exponents. The exponent tells us how many times the base is multiplied by itself.
For example, \( 6^2 \) means that the base 6 is used as a factor twice, which means \( 6 \times 6 = 36 \).

When you see an exponentiation in a problem like the one provided, solve it first before performing any addition or subtraction. This order of operations ensures that the expression is simplified correctly.

• Start with calculating any numbers with exponents.
• Proceed to perform addition or subtraction.

This hierarchy helps in maintaining the correct order of operations, ultimately resulting in an accurate and simplified solution.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator are as small as possible without changing the fraction's value. This often involves finding a common factor between the two numbers.
In our exercise, after evaluating the expressions in both the numerator and denominator, we found that both equal 24, leading to the fraction \( \frac{24}{24} \).

• The fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, in this case, 24.
• As a result, \( \frac{24}{24} \) becomes \( 1 \).

This demonstrates the importance of understanding and applying the concept of simplifying fractions to find the most straightforward form of any expression.