/*! 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 8 For exercises 1-66, simplify. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 8

For exercises 1-66, simplify. $$ \frac{27 h k^{4}}{54 h k} $$

Short Answer

Expert verified
The simplified form is \( \frac{k^3}{2} \).

Step by step solution

01

Identify Common Factors

Look at the numerator and denominator of the fraction. Identify common factors that can be found in both parts. The given fraction is \( \frac{27 hk^4}{54 hk} \). Notice that both the numerator and denominator have the terms 27, 54, h, and k.
02

Simplify Numerical Coefficients

Divide the numerical coefficients in the numerator and denominator by their greatest common divisor (GCD). The GCD of 27 and 54 is 27. Thus \( \frac{27}{54} = \frac{1}{2} \). Now the fraction is \( \frac{1hk^4}{2hk} \).
03

Simplify Variables

Next, simplify the variables. The common variables h and k are in both the numerator and the denominator. Simplify by cancelling like terms: \( \frac{1h k^{4}}{2h k} \) becomes \( \frac{k^3}{2} \) because \( k^4 / k = k^3 \) and \( h/h = 1 \).
04

Final Simplified Expression

The final simplified form of the fraction is \( \frac{k^3}{2} \).

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 fractions, one of the key steps is determining the Greatest Common Divisor (GCD). The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For example, to simplify the fraction \( \frac{27 \, hk^4}{54 \, hk} \), you'll need to find the GCD of 27 and 54.

In this case, the GCD is 27. This means that both 27 and 54 can be divided without leaving a remainder. To simplify the numerical part of the fraction, divide both the numerator and the denominator by their GCD: \[ \frac{27}{54} = \frac{1}{2} \]. This reduces our fraction for numerical coefficients.
Numerical Coefficients
Numerical coefficients are the numbers that multiply the variables in an algebraic expression. In the fraction \( \frac{27 \, hk^4}{54 \, hk} \), 27 and 54 are the numerical coefficients. Once you find their GCD, divide them to simplify the fraction.

We've already seen that dividing 27 and 54 by their GCD (27) yields \[ \frac{27}{54} = \frac{1}{2} \]. After this division, the fraction now looks like \( \frac{1hk^4}{2hk} \). We've simplified the numerical part, making it easier to handle and focus on the variables.
Canceling Like Terms
Canceling like terms is one of the most useful tricks in simplifying fractions. If the same variable appears in both the numerator and the denominator, we can cancel them out. In our fraction \( \frac{27hk^4}{54hk} \), both h and k are in the numerator and the denominator.

To simplify, divide both numerator and denominator by their common terms:
  • For h: \( h/h = 1 \)
  • For k: \( k^4 / k = k^3 \)
This simplifies our fraction to \( \frac{k^3}{2} \). The canceling of like terms eliminates extra variables, making the equation simpler and easier to understand.
Variable Simplification
When simplifying algebraic fractions, focus on reducing the variables. After canceling out like terms, ensure all variables are in their simplest form.

In our example: \( \frac{27hk^4}{54hk} \), by canceling the common terms h and k, the simplified version is \( \frac{k^3}{2} \). Here we reduced \[ k^4 / k = k^3 \], ensuring no unnecessary variables are complicating the expression.

Such variable simplification is essential in algebra as it makes equations more manageable and highlights the core variables influencing the final result.

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

The relationship of the distance driven, \(x\), and the cost of gasoline, \(y\), is a direct variation. For a trip of \(250 \mathrm{mi}\), the cost is \(\$ 90\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost of gasoline to drive \(225 \mathrm{mi}\). d. What does \(k\) represent in this equation?

MRI scans of women with the BRCA1 and BRCA2 genetic mutations that were positive for cancer were wrong five out of six times. (These results are "false positives.") If 1500 women with these mutations had MRI scans that indicated cancer, predict how many of these women did not have cancer. (Source: www.telegraph .co.uk, March 26, 2008)

The relationship of the time a tour guide works, \(x\), and the cost to hire the tour guide, \(y\), is a direct variation. When a tour guide works for \(15 \mathrm{hr}\), the cost is \(\$ 1125\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost to hire a tour guide for \(8 \mathrm{hr}\). d. What does \(k\) represent in this equation?

For a fixed number of hotel rooms, the number of rooms cleaned per hour, \(x\), and the number of hours it takes to clean the rooms, \(y\), is an inverse variation. If a person can clean 8 rooms per hour, it takes 15 hr to clean the rooms. a. Find the constant of variation, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. If a person can clean 6 rooms per hour, find the time needed to clean the rooms.

The relationship of \(x\) and \(y\) is a direct variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\). d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=5\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.