Problem 49 - Prealgebra

Question

Expand and simplify each of the following. $$\left(\frac{3}{4}\right)^{2} \cdot \frac{8}{9}$$

Step-by-Step Solution

Answer

The expanded and simplified expression is \( \frac{1}{2} \).

Step 1: Square the fraction

First, we need to square \( \frac{3}{4} \). This means multiplying \( \frac{3}{4} \) by itself: \[ \left(\frac{3}{4}\right)^2 = \frac{3}{4} \times \frac{3}{4} = \frac{3 \times 3}{4 \times 4} = \frac{9}{16} \]

Step 2: Multiply the squared result by the other fraction

Now, take the squared result \( \frac{9}{16} \) and multiply it by \( \frac{8}{9} \): \[ \frac{9}{16} \times \frac{8}{9} = \frac{9 \times 8}{16 \times 9} = \frac{72}{144} \]

Step 3: Simplify the fraction

To simplify \( \frac{72}{144} \), find the greatest common divisor of 72 and 144, which is 72. Divide both the numerator and the denominator by 72: \[ \frac{72 \div 72}{144 \div 72} = \frac{1}{2} \]

Key Concepts

Squaring Fractions

Squaring a fraction involves multiplying the fraction by itself. For example, if you have the fraction \( \frac{3}{4} \), squaring it means doing the operation \( \frac{3}{4} \times \frac{3}{4} \). This results in a new fraction:

• The numerator is found by multiplying the top parts together: \(3 \times 3 = 9\).
• The denominator comes from multiplying the bottom parts: \(4 \times 4 = 16\).

So, \( \left(\frac{3}{4}\right)^2 = \frac{9}{16} \). Remember that when squaring, you're just repeating the multiplication of the same fraction.

Multiplying Fractions

When you multiply fractions, the process is straightforward. You multiply across the top of the fractions to get a new numerator and across the bottom to get a new denominator. For instance, with \( \frac{9}{16} \times \frac{8}{9} \):

• Multiply the numerators: \(9 \times 8 = 72\).
• Multiply the denominators: \(16 \times 9 = 144\).

This gives you the fraction \( \frac{72}{144} \). Simplifying this result comes next to make it easier to work with.

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with. Simplification involves finding a fraction with the same value but smaller, whole numbers. Begin with \( \frac{72}{144} \). To simplify:

• Identify the greatest common divisor (GCD) of 72 and 144.
• Divide both numerator and denominator by this GCD.

Here, the GCD is 72. So, \( \frac{72}{144} = \frac{72 \div 72}{144 \div 72} = \frac{1}{2} \). This fraction is now in its simplest form.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest whole number that can be evenly divided into two or more numbers. Finding the GCD helps in simplifying fractions. To find the GCD of 72 and 144:

• List the factors of each number.
• Determine which factors are common to both numbers.
• The largest of these common factors is the GCD.

For 72 and 144, their GCD is 72. This means both numbers can be divided by 72 without leaving a remainder, helping simplify fractions like \( \frac{72}{144} \) to \( \frac{1}{2} \).