Problem 78 - Elementary Algebra

Question

\(\left(\frac{3}{4}+\frac{1}{2}\right)^{2}\)

Step-by-Step Solution

Answer

\(\frac{25}{16}\)

Step 1: Add the fractions

First, find a common denominator for the fractions \(\frac{3}{4}\) and \(\frac{1}{2}\). The least common denominator is 4. Convert \(\frac{1}{2}\) to \(\frac{2}{4}\) and add the fractions: \(\frac{3}{4} + \frac{2}{4} = \frac{5}{4}\).

Step 2: Square the result

Next, square \(\frac{5}{4}\): \(\frac{5}{4}\) squared is \(\frac{5}{4} \times \frac{5}{4} = \frac{25}{16}\).

Key Concepts

Adding Fractions

To add fractions, you need to make sure that the denominators (the bottom numbers) are the same. This is because fractions need a common base to be added properly. For instance, when adding \(\frac{3}{4}\) and \(\frac{1}{2}\), you first need a common denominator. The smallest common denominator for 4 and 2 is 4.
Therefore, convert \(\frac{1}{2}\) to \(\frac{2}{4}\). Once the fractions have the same denominator, you can add the numerators (the top numbers) directly. So, \(\frac{3}{4} + \frac{2}{4} = \frac{5}{4}\).
Remember these key steps while adding fractions:

• Find a common denominator.
• Convert fractions to have the same denominator, if necessary.
• Add the numerators while keeping the same denominator.

This is crucial in ensuring that the fractions are added correctly.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. The smallest common multiple is called the Least Common Denominator (LCD). Finding the LCD is a crucial step when adding, subtracting, or comparing fractions.
For example, in adding \(\frac{3}{4}\) and \(\frac{1}{2}\), you will notice that their denominators (4 and 2) need to match. The LCD for 4 and 2 is 4 because 4 is the smallest number that both 4 and 2 can be multiplied into evenly.
Converting \(\frac{1}{2}\) into \(\frac{2}{4}\) aligns its denominator with \(\frac{3}{4}\), making it possible to add the fractions easily.
Steps to find a common denominator:

• Identify the denominators of the fractions.
• Find the Least Common Multiple (LCM) of these denominators.
• Convert each fraction to an equivalent fraction with the LCD as the new denominator.

This enables smooth addition and subtraction of fractions.

Squaring Fractions

Squaring a fraction means multiplying the fraction by itself. This involves both the numerator and denominator. For example, to square \(\frac{5}{4}\), you multiply 5 by 5 and 4 by 4. This results in \(\frac{25}{16}\).
When squaring fractions, follow these steps:

• Square the numerator (top number).
• Square the denominator (bottom number).
• Simplify the resulting fraction if possible.

Applying these steps to our example:
\(\frac{5}{4} \times \frac{5}{4} = \frac{25}{16}\).
Note that the fraction \(\frac{25}{16}\) cannot be simplified further, and this will be your final answer.