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Chapter 1: Problem 21

$$2 \cdot \frac{3}{4}-\left(\frac{2}{3}\right)^{2}$$

Short Answer

Expert verified
The short version of the answer is: \[\frac{19}{18}\]

Step by step solution

01

Deal with exponents inside the parentheses

First, we need to deal with the exponent inside the parentheses, which is to calculate \(\left(\frac{2}{3}\right)^{2}\). To do this, we square both the numerator and the denominator: \[\left(\frac{2}{3}\right)^{2} = \frac{(2)^{2}}{(3)^{2}} = \frac{4}{9}\] Now our expression becomes: \[2 \cdot \frac{3}{4} - \frac{4}{9}\]
02

Simplify the multiplication

Next, we will handle the multiplication part of the expression: \[2 \cdot \frac{3}{4}\] When multiplying a whole number by a fraction, we can write the whole number as a fraction with a denominator of 1 and proceed with multiplication: \[\frac{2}{1} \cdot \frac{3}{4}\] Now, multiply the numerators and denominators separately: \[\frac{(2)(3)}{(1)(4)} = \frac{6}{4}\] Now, our expression becomes: \[\frac{6}{4} - \frac{4}{9}\]
03

Find a common denominator and subtract

Before we can subtract the two fractions, we must find a common denominator. In this case, the least common multiple (LCM) of 4 and 9 is 36. Now, we need to convert both fractions to equivalent fractions with a denominator of 36 by multiplying their numerators and denominators by the appropriate factors. For \(\frac{6}{4}\): \[\frac{6}{4} \cdot \frac{9}{9} = \frac{54}{36}\] For \(\frac{4}{9}\): \[\frac{4}{9} \cdot \frac{4}{4} = \frac{16}{36}\] Now, our expression becomes: \[\frac{54}{36} - \frac{16}{36}\] Finally, subtract the numerators and keep the denominator the same: \[\frac{54 - 16}{36} = \frac{38}{36}\]
04

Simplify the final fraction

To simplify the final fraction, we find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 38 and 36 is 2. Now, divide both the numerator and the denominator by the GCD: \[\frac{38}{36} = \frac{38 \div 2}{36 \div 2} = \frac{19}{18}\] Therefore, the simplified answer is \[\frac{19}{18}\]

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Most popular questions from this chapter

Objectives 3 and 4 Clarice's parents tell her that she must deposit \(\frac{1}{3}\) of the money she earns from babysitting into her savings account, but she can keep the rest. If she earns \(\$ 117\) in 1 week during the summer how much does she deposit, and how much does she keep?

Rewrite each expression using the distributive property. Simplify if possible. $$-(x-4 y+10 z)$$

Write a mathematical expression for each and simplify. 19 less than the product of \(-4\) and \(-8\)

Rewrite each expression using the distributive property. Simplify if possible. $$-8(u-4)$$

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