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Chapter 3: Problem 49

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

Short Answer

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

Step by step solution

01

Square the fraction

First, we need to square \( \frac{3}{4} \). This means multiplying \( \frac{3}{4} \) by itself: \[ \left(\frac{3}{4}\right)^2 = \frac{3}{4} \times \frac{3}{4} = \frac{3 \times 3}{4 \times 4} = \frac{9}{16} \]
02

Multiply the squared result by the other fraction

Now, take the squared result \( \frac{9}{16} \) and multiply it by \( \frac{8}{9} \): \[ \frac{9}{16} \times \frac{8}{9} = \frac{9 \times 8}{16 \times 9} = \frac{72}{144} \]
03

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: \[ \frac{72 \div 72}{144 \div 72} = \frac{1}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \( \frac{1}{2} \).

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Most popular questions from this chapter

Use the rule for order of operations to combine the following. \(9 \cdot 7+6 \cdot 5\)

Multiply or divide as indicated. $$12 \cdot \frac{3}{4}$$

Multiply or divide as indicated. $$4 \cdot \frac{3}{4}$$

Multiply or divide as indicated. $$\frac{11}{12} \cdot \frac{10}{11} \cdot \frac{9}{10} \cdot \frac{8}{9} \cdot \frac{7}{8}$$

Find the following sums. (Add.) $$\begin{array}{r}1 \frac{5}{6} \\\2 \frac{3}{4} \\\\+5 \frac{1}{2} \\\\\hline\end{array}$$
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