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Chapter 4: Problem 696

In the following exercises, simplify. $$ \frac{2^{3}-2^{2}}{\left(\frac{3}{4}\right)^{2}} $$

Short Answer

Expert verified
The simplified expression is \( \frac{64}{9} \).

Step by step solution

01

Simplify the Numerator

Simplify the numerator by calculating each term separately. We have two terms: \( 2^3 \) and \( 2^2 \). Calculate: \( 2^3 = 8 \) and \( 2^2 = 4 \). Now subtract: \( 8 - 4 = 4 \). Thus, the numerator simplifies to 4.
02

Simplify the Denominator

Simplify the denominator by calculating \( \left( \frac{3}{4} \right)^2 \). Square both the numerator and the denominator separately: \( \left( \frac{3}{4} \right)^2 = \frac{3^2}{4^2} = \frac{9}{16} \). Thus, the denominator simplifies to \( \frac{9}{16} \).
03

Divide Numerator by Denominator

Now divide the simplified numerator by the simplified denominator: \( \frac{4}{\frac{9}{16}} \). Dividing by a fraction is equivalent to multiplying by its reciprocal: \( 4 \times \frac{16}{9} \).
04

Multiply and Simplify

Multiply the terms: \( 4 \times \frac{16}{9} = \frac{4 \times 16}{9} = \frac{64}{9} \). Thus, the expression simplifies to \( \frac{64}{9} \).

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Key Concepts

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

Exponents
Exponents represent repeated multiplication of a number by itself. For example, in the expression given in the exercise, we have two terms: 2^3 and 2^2. Let's break them down:

  • 2^3 means multiplying 2 by itself three times: 2 * 2 * 2 = 8.
  • 2^2 means multiplying 2 by itself twice: 2 * 2 = 4.

Exponents make calculations easier by providing a shorthand for multiplication. When simplifying expressions, always calculate the exponents first as part of the order of operations.
Fractions
Fractions are a way of representing parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). In our exercise, the fraction we dealt with was \(\left(\frac{3}{4}\right)^2\). Here鈥檚 a step-by-step explanation:

  • When we square a fraction, we square both the numerator and the denominator. So, \(\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2}\).
  • Calculating the squares: \(3^2 = 9\) and \(4^2 = 16\).
  • This means \(\left(\frac{3}{4}\right)^2 = \frac{9}{16}\).

When dividing fractions, remember we invert (flip) the divisor and multiply. For instance, instead of dividing by \(\frac{9}{16}\), we multiply by \(\frac{16}{9}\).
Order of Operations
The order of operations is a set of rules that determines the sequence in which calculations are performed. Following these rules ensures that everyone gets the same result. Here are the basic steps, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction):

  • Parentheses: Complete any arithmetic inside parentheses first.
  • Exponents: Next, solve any exponents (powers and roots).
  • Multiplication and Division: Work these out from left to right.
  • Addition and Subtraction: Complete these operations last, from left to right.

In our exercise:
  1. First, we calculated the exponents in the numerator and denominator: \(2^3 - 2^2\) and \(\left(\frac{3}{4}\right)^2\).
  2. Second, we simplified those results: \(8 - 4\) for the numerator and \(\frac{9}{16}\) for the denominator.
  3. Finally, we divided the simplified numerator by the simplified denominator, resulting in \(\frac{64}{9}\).

By following the order of operations, you ensure every step is completed correctly and that the final result is accurate.

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Most popular questions from this chapter

In the following exercises, simplify. $$\left(\frac{5}{9}+\frac{1}{6}\right) \div\left(\frac{2}{3}-\frac{1}{2}\right)$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{5}{12}-w\) when (a) \(w=\frac{1}{4} \quad\) (b) \(w=-\frac{1}{4}\)

In the following exercises, solve the equation. $$ f+\left(-\frac{2}{3}\right)=\frac{5}{12} $$

In the following exercises, simplify. $$ -\frac{3}{4} \div \frac{x}{3} $$

In the following exercises, simplify. $$\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}$$
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