/*! 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 4 For exercises 1-66, simplify. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 4

For exercises 1-66, simplify. $$ \frac{54 c^{2} d^{5}}{72 c d} $$

Short Answer

Expert verified
The simplified expression is \ \ \ \[ \frac{3c d^{4}}{4} \]

Step by step solution

01

- Write Down the Original Expression

The given expression is \ \ \ \[ \frac{54 c^{2} d^{5}}{72 c d} \]
02

- Simplify the Numerical Coefficients

Reduce the numerical part of the fraction. Both 54 and 72 can be divided by their greatest common divisor, which is 18: \ \ \ \[ \frac{54}{72} = \frac{54 \div 18}{72 \div 18} = \frac{3}{4} \] \ Therefore, the expression simplifies to: \ \ \ \[ \frac{3 c^{2} d^{5}}{4 c d} \]
03

- Simplify the Variables

Next, simplify the variables by canceling common terms in the numerator and the denominator. For the variable 'c', we have \ \ \ \[ c^{2} \div c = c^{2-1} = c^{1} = c \] \ and for the variable 'd', we have: \ \ \ \[ d^{5} \div d = d^{5-1} = d^{4} \] \ Therefore, the simplified expression becomes: \ \ \ \[ \frac{3c d^{4}}{4} \]
04

- Final Simplified Expression

Combine all the simplified parts. The completely simplified expression is: \ \ \ \[ \frac{3c d^{4}}{4} \]

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.

Greatest Common Divisor
When simplifying algebraic fractions, the first step is to reduce the numerical coefficients. To do this effectively, we need to identify their Greatest Common Divisor (GCD). The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
For the expression \( \frac{54 c^{2} d^{5}}{72 c d} \), we start by reducing the numbers 54 and 72.
  • 54 can be factored into \( 2 \times 3^3 \)
  • 72 can be factored into \( 2^3 \times 3^2 \)

Both numbers share common factors of 2 and 3. The largest combination of these shared factors is 18. Therefore, the GCD of 54 and 72 is 18.
We can now divide the numerator and denominator by 18:
  • \( \frac{54}{18} = 3 \)
  • \( \frac{72}{18} = 4 \)

This simplifies our fraction to \( \frac{3 c^{2} d^{5}}{4 c d} \).
Exponent Rules
Exponents are shorthand for repeated multiplication, and there are specific rules for simplifying them. Let's review the key rules:
  • \textbf{Product of Powers Rule:} \( a^m \times a^n = a^{m+n} \)
  • \textbf{Quotient of Powers Rule:} \( \frac{a^m}{a^n} = a^{m-n} \) (when \( n < m \))
  • \textbf{Power of a Power Rule:} \( (a^m)^n = a^{mn} \)

In our problem, the Quotient of Powers Rule is particularly useful. Our fraction now is \( \frac{3 c^{2} d^{5}}{4 c d} \).
We need to simplify the variables by applying the rule:
  • For variable \( c \): \( c^{2-1} = c^1 = c \)
  • For variable \( d \): \( d^{5-1} = d^4 \)

By reducing the exponents, our fraction becomes \( \frac{3 c d^4}{4} \).
Variable Simplification
Variable simplification involves reducing the expression by canceling out common terms.
After simplifying the coefficients and exponents, we have the fraction \( \frac{3 c^{2} d^{5}}{4 c d} \).
Cancel out the common variable terms in both numerator and denominator:
  • For \( c \), the numerator has \( c^2 \), and the denominator has \( c \). By applying the quotient rule for exponents, we get \( c^{2-1} = c^1 = c \).
  • For \( d \), the numerator has \( d^5 \), and the denominator has \( d \). Applying the quotient rule, we get \( d^{5-1} = d^4 \).

Now, our fraction looks like \( \frac{3c d^4}{4} \).
Hence, the final simplified expression for the given algebraic fraction is \( \frac{3c d^4}{4} \).

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

Find the number of 80,500 children and adolescents who are obese. Round to the nearest hundred. Approximately one in six children and adolescents are obese. (Source: www.cdc.gov, Nov. 2011)

If the annual credit sales are constant, the relationship of the accounts receivable, \(x\), and the financial ratio receivables turnover, \(y\), is an inverse variation. The accounts receivable of a company are \(\$ 150\) million, and its receivables turnover ratio is 12 . a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the receivables turnover ratio when the accounts receivable are \(\$ 200\) million.

For exercises \(67-82\), use the five steps and a proportion. A survey asked 505 companies whether they would continue to match their employees' contributions to their \(401 \mathrm{k}\) retirement plans. Find the number of companies that will continue to match the contributions. Three out of five employers maintain \(401(\mathrm{k})\) match despite economic crisis. (Source: www.americanbenefitscouncil.org, March 17, 2009)

For exercises \(67-82\), use the five steps and a proportion. Find the number of adults used to create the ratio "four out of five." Four out of five adults now use the Internet. 184 million adults are online from their homes, offices, schools or other locations. (Source: www.harrisinteractive.com, Nov. 17, 2008)

The relationship of the number of tickets sold, \(x\), and the total ticket receipts for an outdoor concert, \(y\), is a direct variation. When 11,000 tickets are sold, the total ticket receipts are \(\$ 495,000\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the number of tickets sold when the total ticket receipts are \(\$ 562,500\). d. Find the total ticket receipts from the sale of 7575 tickets. e. What does \(k\) represent in this equation?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.