/*! 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 290 Divide the monomials. $$ \fr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 290

Divide the monomials. $$ \frac{18 a^{4} b^{8}}{-27 a^{9} b^{5}} $$

Short Answer

Expert verified
-\frac{2 b^{3}}{3 a^{5}}

Step by step solution

01

Simplify the coefficients

Divide the numerical coefficients: \( \frac{18}{-27} = -\frac{2}{3} \).
02

Apply the quotient rule for exponents (a)

For the variable \( a \on\frac{a^{4}}{a^{9}}: a^{4-9} = a^{-5} \).
03

Apply the quotient rule for exponents (b)

For the variable \( b \on\frac{b^{8}}{b^{5}}: b^{8-5} = b^{3} \).
04

Combine the simplified parts

Combine the results from previous steps: \(-\frac{2}{3} a^{-5} b^{3} \).
05

Rewrite using positive exponents

Rewrite \( a^{-5} \) as \( \frac{1}{a^{5}} \). So, the expression becomes \(-\frac{2 b^{3}}{3 a^{5}} \).

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.

quotient rule for exponents
When you divide monomials with the same base, you use the quotient rule for exponents. This rule states that you subtract the exponent of the denominator from the exponent of the numerator.

For example, consider the division \( \frac{a^m}{a^n} \). According to the quotient rule, this can be simplified to: \[ a^{m-n} \].

In our exercise, we had \( \frac{a^{4}}{a^{9}} \). Using the quotient rule results in: \[ a^{4-9} = a^{-5} \].

Similarly, for the variable \( b \), we had \( \frac{b^{8}}{b^{5}} \), which simplifies to: \[ b^{8-5} = b^{3} \].

Remember, this rule only applies to terms with the same base. The bases do not change; you only adjust the exponents.
simplify coefficients
Simplifying the coefficients is the first step in dividing monomials. Coefficients are the numerical parts of the terms being divided. Follow these easy steps:

  • Take the numerical coefficient from the numerator and the denominator.
  • Divide them as you would with any two numbers.


In our exercise, the coefficients were 18 and -27. To simplify this: \[ \frac{18}{-27} = -\frac{2}{3} \].

A negative result means that the entire fraction will be negative. Do not ignore negative signs! They are crucial to the accuracy of your solution.
positive exponents
When working with exponents, it's best to write your final answers with positive exponents. This makes your solution easier to understand and more universally accepted.

If you end up with a negative exponent, rewrite it using positive exponents. Use this rule: \[ a^{-n} = \frac{1}{a^n} \].

In our exercise, we got \( a^{-5} \) which we converted to a positive exponent as: \[ \frac{1}{a^5} \].

By combining the simplified coefficient and using positive exponents, our final result became: \( - \frac{2 b^{3}}{3 a^{5}} \).

Always aim to express your terms with positive exponents for neatness and clarity.

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 each product. $$ \left(r^{2}-s^{3}\right)\left(r^{3}+s^{2}\right) $$

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Multiply. (a) \(-8 x\left(x^{2}+2 x-15\right)\) (b) \(5 p q^{3}\left(p^{2}-2 p q+6 q^{2}\right)\)

Multiply the binomials. Use any method. $$ (2 t-9)(10 t+1) $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (5 k+6)(5 k-6) $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics 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.