Problem 48 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(\frac{3}{8}\right)^{2}+\frac{1}{4}+\frac{1}{8} \cdot \frac{3}{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{37}{64} \).

Step 1: Square the fraction

Start by simplifying the first term, \( \left( \frac{3}{8} \right)^2 \). To do this, square both the numerator and the denominator: \[ (3)^2 = 9 \quad \text{and} \quad (8)^2 = 64 \] Thus, \[ \left( \frac{3}{8} \right)^2 = \frac{9}{64} \]

Step 2: Convert fraction to like denominator

Next, look at the second term, \( \frac{1}{4} \). To add this to \( \frac{9}{64} \), we need a common denominator. Since the denominator of \( \frac{9}{64} \) is 64, convert \( \frac{1}{4} \) to this same denominator: \[ \frac{1}{4} = \frac{1 \times 16}{4 \times 16} = \frac{16}{64} \]

Step 3: Multiply fractions

Now, look at the third term, \( \frac{1}{8} \cdot \frac{3}{2} \). Multiply the numerators and the denominators: \[ \frac{1}{8} \cdot \frac{3}{2} = \frac{1 \cdot 3}{8 \cdot 2} = \frac{3}{16} \] We need this result as a fraction with the same denominator as the others. Convert \( \frac{3}{16} \) to \( \frac{64}{64} \) by multiplying both the numerator and the denominator by 4: \[ \frac{3}{16} = \frac{3 \times 4}{16 \times 4} = \frac{12}{64} \]

Step 4: Add fractions

Add all the fractions together, \( \frac{9}{64} + \frac{16}{64} + \frac{12}{64} \): Combine the numerators: \[ 9 + 16 + 12 = 37 \] Thus, \[ \frac{9}{64} + \frac{16}{64} + \frac{12}{64} = \frac{37}{64} \]

Step 5: Simplify the final result

Check if the fraction \( \frac{37}{64} \) can be simplified further. Since 37 is a prime number and does not divide 64 evenly, the fraction is already in its simplest form.

Key Concepts

Squaring Fractions

When we talk about squaring a fraction, it simply means multiplying the fraction by itself. This involves squaring both the numerator (the top number) and the denominator (the bottom number).
For example, if we have the fraction \(\frac{3}{8}\), we square it by doing the following steps:

• Square the numerator: \(3^2 = 9\).
• Square the denominator: \(8^2 = 64\).

Thus, \(\left(\frac{3}{8}\right)^2 = \frac{9}{64}\). This fraction \(\frac{9}{64}\) remains its simplest form as it cannot be reduced further. Each part of the fraction, when squared, becomes larger, which is typical of squaring operations, whether the base is a fraction or a whole number.

Addition of Fractions

Adding fractions can be a bit tricky, especially when their denominators are different. To add fractions, they must share a common denominator. This may require converting fractions to have like denominators before carrying out the addition.
Consider two fractions, \(\frac{9}{64}\) and \(\frac{16}{64}\), which already share a common denominator: this means they can be added directly by combining their numerators:

• Add their numerators: \(9 + 16 = 25\).

Therefore, \(\frac{9}{64} + \frac{16}{64} = \frac{25}{64}\). However, fractions with unlike denominators require additional steps to convert to a common denominator before adding their numerators.

Common Denominator

Achieving a common denominator for fractions is crucial for operations like addition and subtraction. The common denominator is the smallest number that is a multiple of the denominators of all fractions involved.
In the provided expression, two fractions are \(\frac{1}{4}\) and \(\frac{9}{64}\). The denominator 4 needs to be converted to 64, which is found by multiplying by 16:

• Convert \(\frac{1}{4}\) to \(\frac{16}{64}\): \(\frac{1 \times 16}{4 \times 16}\).

This allows for easy addition or subtraction. A common denominator ensures fractions are comparable, making calculations straightforward.

Multiplication of Fractions

Multiplying fractions involves a simple process: multiply their numerators together and their denominators together. You do not need a common denominator to multiply fractions.
For instance, consider \(\frac{1}{8} \cdot \frac{3}{2}\). Multiply the numerators and the denominators:

• For the numerators, \(1 \times 3 = 3\).
• For the denominators, \(8 \times 2 = 16\).

Thus, \(\frac{1}{8} \cdot \frac{3}{2} = \frac{3}{16}\). This fraction can then be altered to share a common denominator with others if needed. Here, it was converted to \(\frac{12}{64}\) by multiplying both the numerator and denominator by 4, aligning with a common denominator for subsequent operations.