/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 201 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 201

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

Short Answer

Expert verified
54

Step by step solution

01

- Simplify the numerator

First, simplify the expression in the numerator: \(2^3 + 4^2\). \(2^3 = 2 \times 2 \times 2 = 8\). \(4^2 = 4 \times 4 = 16\). So, the numerator becomes \(8 + 16 = 24\).
02

- Simplify the denominator

Now, simplify the denominator: \(\left(\frac{2}{3}\right)^2\). To square a fraction, square both the numerator and the denominator: \(\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}\).
03

- Divide the simplified numerator by the simplified denominator

Next, divide the numerator by the denominator: \(\frac{24}{\frac{4}{9}}\). This is the same as multiplying by the reciprocal of the denominator: \(24 \times \frac{9}{4}\).
04

- Multiply and simplify

Finally, compute the multiplication: \(24 \times \frac{9}{4}\). First, multiply the numerators: \(24 \times 9 = 216\). Then, divide by the denominator: \(\frac{216}{4} = 54\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, in the step-by-step solution, we see expressions like \(2^3\) and \(4^2\). The exponent tells us how many times to multiply the base number by itself.
  • \(2^3\) means \(2 \times 2 \times 2 = 8\).
  • \(4^2\) means \(4 \times 4 = 16\).
These skills are crucial for simplifying algebraic expressions that contain powers and exponents. Always handle exponents first before moving on to addition or subtraction.
Working with Fractions
Fractions represent parts of a whole. In the problem, we simplified the denominator by squaring the fraction \(\left(\frac{2}{3}\right)^2\).
When squaring a fraction:
  • Square the numerator: \(2^2 = 4\).
  • Square the denominator: \(3^2 = 9\).
Therefore, \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\). Understanding how to manipulate fractions is key to solving more complex problems, especially when they are part of larger expressions.
Integer Operations
Integer operations involve basic arithmetic functions, such as addition, subtraction, multiplication, and division, with whole numbers. In the solution, the integer operations are:
  • Addition: In the numerator, \(8 + 16 = 24\).
  • Multiplication: \(24 \times 9 = 216\).
  • Division: \(\frac{216}{4} = 54\).
These steps demonstrate how arithmetic rules apply when dealing with integers in algebraic expressions. Each operation must be performed in the correct order to simplify the expression properly.
Understanding Reciprocals
A reciprocal is simply the inverse of a number or fraction. It's what you multiply by to get 1. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
In the solution, the reciprocal comes into play when dividing fractions:
  • To divide by a fraction, multiply by its reciprocal.
  • For \(\frac{24}{\frac{4}{9}}\), multiply by the reciprocal of \(\frac{4}{9}\), which is \(\frac{9}{4}\).
Therefore, \(24 \times \frac{9}{4}\) simplifies to \(\frac{216}{4} = 54\). Understanding reciprocals helps greatly in performing operations involving divisions of fractions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In the following exercises, list the (a) whole numbers, (b) integers, \(\odot\) rational numbers, \(@\) irrational numbers, \(\Theta\) real numbers for each set of numbers. $$ -8,0,1.95286 \ldots, \frac{12}{5}, \sqrt{36}, 9 $$

In the following exercises, simplify. $$ \sqrt{169} $$

In the following exercises, simplify using the Distributive Property. $$ 5(2 n+9)+12(n-3) $$

In your own words, state the Associative Property of addition.

In the following exercises, simplify. $$ \frac{2}{5}+\frac{5}{12}+\left(-\frac{2}{5}\right) $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.