Problem 102 - Elementary Algebra

Question

For exercises 97-114, evaluate. $$ \frac{8}{9}+\left(\frac{1}{2}-\frac{1}{6}\right)^{2} $$

Step-by-Step Solution

Answer

1

Step 1 - Simplify the Expression in Parentheses

First, simplify the expression inside the parentheses: \ \( \frac{1}{2} - \frac{1}{6} \). To do this, find a common denominator. The common denominator of 2 and 6 is 6: \ \( \frac{1}{2} = \frac{3}{6} \). So, \ \( \frac{1}{2} - \frac{1}{6} = \frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \).

Step 2 - Square the Simplified Expression

Now square the simplified expression to remove the parentheses: \ \( \left( \frac{1}{3} \right)^{2} \ = \frac{1}{9} \).

Step 3 - Add the Result to the First Fraction

Add this squared result to the first fraction in the original problem: \ \( \frac{8}{9} + \frac{1}{9} = \frac{8 + 1}{9} = \frac{9}{9} = 1 \).

Key Concepts

Common Denominator

When dealing with fractions, finding a common denominator is often necessary. This is especially true for addition and subtraction. A common denominator is a shared multiple of the denominators in the fractions you're working with.

For example, in the expression \(\frac{1}{2} - \frac{1}{6}\), the denominators are 2 and 6. To find a common denominator, you can list the multiples of each denominator.

• Multiples of 2: 2, 4, 6, 8, 10, ...
• Multiples of 6: 6, 12, 18, 24, ...

You can see that 6 is the smallest multiple that both 2 and 6 share, so it's our common denominator.

Next, convert each fraction to have this common denominator:

• \(\frac{1}{2} = \frac{3}{6}\)
• \(\frac{1}{6}\) remains the same

Now you can easily subtract: \(\frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\).

Squaring Fractions

Squaring a fraction is very straightforward. To square a fraction, you multiply the numerator by itself and the denominator by itself. This essentially means you square the fraction's numerator and denominator separately.

For instance, to square \(\frac{1}{3}\), you do the following:

• Square the numerator: \(1^{2} = 1\)
• Square the denominator: \(3^{2} = 9\)

Therefore, \(\frac{1}{3}^{2}\) becomes \(\frac{1}{9}\).

In our problem, once we simplified \(\frac{1}{2} - \frac{1}{6}\) to get \(\frac{1}{3}\), we then squared \(\frac{1}{3}\) and got \(\frac{1}{9}\).

Fraction Addition

Adding fractions requires a common denominator, just like subtraction. Once all fractions have the same denominator, you can simply add the numerators together.

In our exercise, we needed to add \( \frac{8}{9} + \frac{1}{9} \). Since both fractions already have a common denominator, 9, we just add the numerators:

\( 8 + 1 = 9 \)

So, \( \frac{8}{9} + \frac{1}{9} = \frac{9}{9} = 1 \).

Important tips for adding fractions:

• The denominators must be the same
• Only the numerators are added
• Reduce the resulting fraction if possible

Following these steps ensures accurate fraction addition.