/*! 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 78 Perform the indicated operation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 78

Perform the indicated operation. If possible, simplify your answer. $$ \left(\frac{2 x}{3}\right)^{2} \cdot\left(\frac{3}{x}\right)^{2} $$

Short Answer

Expert verified
The simplified result is 4.

Step by step solution

01

Apply the Power of a Quotient

First, apply the power to both the numerator and the denominator separately in both expressions. This means we will square each part. For \(\left(\frac{2x}{3}\right)^2\), this becomes \(\frac{(2x)^2}{3^2}\). For \(\left(\frac{3}{x}\right)^2\), this becomes \(\frac{3^2}{x^2}\).
02

Simplify Each Squared Expression

Calculate each squared term. \((2x)^2 = 4x^2\) and \(3^2 = 9\), so \(\left(\frac{2x}{3}\right)^2 = \frac{4x^2}{9}\). Similarly, \(3^2 = 9\) and \(x^2 = x^2\), resulting in \(\left(\frac{3}{x}\right)^2 = \frac{9}{x^2}\).
03

Multiply the Fractions

To multiply the fractions \(\frac{4x^2}{9}\) and \(\frac{9}{x^2}\), you multiply the numerators together and the denominators together. This results in \(\frac{4x^2 \cdot 9}{9 \cdot x^2} = \frac{36x^2}{9x^2}\).
04

Simplify the Resulting Fraction

The fraction \(\frac{36x^2}{9x^2}\) can be simplified by dividing both the numerator and the denominator by \(9x^2\), resulting in \(\frac{36}{9} = 4\). Therefore, the simplified result is 4.

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.

Power of a Quotient
When you encounter an expression like \( \left( \frac{2x}{3} \right)^2 \), it involves the concept called power of a quotient. This rule allows us to simplify expressions where a fraction is raised to a power. The idea is to distribute the exponent to both the numerator and the denominator separately. For instance, using the power of quotient rule:
  • \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \), where both \(a\) and \(b\) are raised to the same power \(n\).

Applying this to \( \left( \frac{2x}{3} \right)^2 \), we get \( \frac{(2x)^2}{3^2} \). Breaking it down:
  • The expression \((2x)^2\) means that each part, 2 and \(x\), is squared separately, yielding \(4x^2\).
  • The term \(3^2\) is simply \(9\).

This operation simplifies complex algebraic expressions and is a crucial skill in algebra.
Simplifying Fractions
Simplifying fractions is about reducing them to their most basic form, where the numerator and denominator have no common factors other than 1. For the expression \( \frac{36x^2}{9x^2} \), simplification involves cancelling out the common terms and factors from the numerator and the denominator. Here鈥檚 how to perform this step effectively:
  • Break down \(36x^2\) as \(9x^2 \times 4\).
  • Factor both numerator and denominator, here \(9x^2\) appears in both.

  • Cancel the common factor \(9x^2\) from both, which leaves you with \( \frac{4}{1} \), or simply \(4\).

The goal of simplifying is to obtain an expression that's easier to understand, compare, and use in further mathematical operations. Ensure that when simplifying, you follow the rules of arithmetic to prevent mistakes.
Multiplying Fractions
Multiplying fractions is straightforward compared to addition or subtraction. When multiplying, you simply multiply the numerators together and the denominators together, forming a new fraction.
  • For example, \( \frac{4x^2}{9} \times \frac{9}{x^2} \) results in \( \frac{4x^2 \times 9}{9 \times x^2} \).
  • This is \( \frac{36x^2}{9x^2} \) after calculating each product.

The beauty of multiplying fractions is that you can typically simplify before you calculate, which makes the arithmetic easier and faster. It's a great technique for managing larger numbers or more complex expressions effectively. Remember, once you multiply and get a result, always check if it can be simplified further to ensure the answer is in its simplest form.

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

Factor the following. $$ x^{3}-9 x $$

For the given polynomial \(P(x)\) and the given \(c,\) use the remainder theorem to find \(P(c)\). $$ P(x)=2 x^{4}-3 x^{2}-2 ; \frac{1}{3} $$

Find each square root. See Section 1.3. $$ \sqrt{\frac{25}{121}} $$

Explain an advantage of using synthetic division instead of long division.

Kraft Foods is a provider of many of the best-known food brands in our supermarkets. Among their wellknown brands are Kraft, Oscar Mayer, Maxwell House, and Oreo. Kraft Foods' annual revenues since 2005 can be modeled by the polynomial function \(R(x)=0.06 x^{3}+0.02 x^{2}+1.67 x+32.33,\) where \(R(x)\) is revenue in billions of dollars and \(x\) is the number of years since \(2005 .\) Kraft Foods' net profit can be modeled by the function \(P(x)=0.07 x^{3}-0.42 x^{2}+0.7 x+2.63,\) where \(P(x)\) is the net profit in billions of dollars and \(x\) is the number of years since \(2005 .\) (Source: Based on information from Kraft Foods) a. Suppose that a market analyst has found the model \(P(x)\) and another analyst at the same firm has found the model \(R(x) .\) The analysts have been asked by their manager to work together to find a model for Kraft Foods' profit margin. The analysts know that a company's profit margin is the ratio of its profit to its revenue. Describe how these two analysts could collaborate to find a function \(m(x)\) that models Kraft Foods' net profit margin based on the work they have done independently. b. Without actually finding \(m(x),\) give a general description of what you would expect the answer to be.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure 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.