/*! 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 Simplify the square root express... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 8

Simplify the square root expressions. $$ \frac{\sqrt{a^{3 m-5}}}{\sqrt{a^{m-1}}} $$

Short Answer

Expert verified
Question: Simplify the expression \(\frac{\sqrt{a^{3m-5}}}{\sqrt{a^{m-1}}}\). Answer: \(\sqrt{a^{2m-4}}\)

Step by step solution

01

Combine the square root expressions

We can combine the square roots into one expression using the following property: \(\sqrt{a}/\sqrt{b}=\sqrt{a/b}\). Apply this property to the given expression: $$ \frac{\sqrt{a^{3 m-5}}}{\sqrt{a^{m-1}}}=\sqrt{\frac{a^{3m-5}}{a^{m-1}}} $$
02

Simplify the expression under the square root using the properties of exponents

For a given base, use the following property of exponents: \(\frac{a^{n}}{a^{m}}=a^{n-m}\). Apply this property to the expression under the square root: $$ \sqrt{\frac{a^{3m-5}}{a^{m-1}}}=\sqrt{a^{(3m-5)-(m-1)}} $$
03

Simplify the exponent

Carry out the subtraction in the exponent: $$ \sqrt{a^{(3m-5)-(m-1)}}=\sqrt{a^{3m-5-m+1}} $$ Combine the like terms: $$ \sqrt{a^{3m-5-m+1}}=\sqrt{a^{2m-4}} $$
04

Final answer

The simplified expression is: $$ \frac{\sqrt{a^{3m-5}}}{\sqrt{a^{m-1}}}=\sqrt{a^{2m-4}} $$

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.

Combining Square Roots
When working with square roots in algebra, there are several processes that can help simplify complex expressions.
One key technique involves combining multiple square roots into a single square root expression.

In the given problem, we have the expression \(\frac{\sqrt{a^{3m-5}}}{\sqrt{a^{m-1}}}\). By using the property \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), we can combine the two separate square roots into one:

  • This step simplifies the division of the individual roots into a single root of a quotient.
  • It lays the groundwork for further simplification of the expression inside the square root.
Combining square roots is helpful because it allows us to manage and reduce the complexity of an expression at an early stage in the simplification process.
Properties of Exponents
Understanding the properties of exponents is crucial when simplifying expressions that involve powers.
The fundamental rule used in this exercise is the quotient of powers property, which states that \(\frac{a^{n}}{a^{m}}=a^{n-m}\).

Let's apply this to the expression under the square root:
\[\sqrt{\frac{a^{3m-5}}{a^{m-1}}} = \sqrt{a^{(3m-5) - (m-1)}}\]

  • This reduction happens because the same base \(a\) is present in both the numerator and denominator.
  • The exponents undergo subtraction, thus simplifying the power inside the square root.
By organizing exponents in this manner, we can make a more streamlined expression that is closer to the final simplified form.
Algebraic Simplification
Algebraic simplification involves reducing expressions to their simplest form.
After applying the properties of exponents, it is important to carry out any outstanding arithmetic to finalize the simplification.

In the exercise, the expression \(\sqrt{a^{(3m-5)-(m-1)}}\) was transformed to \(\sqrt{a^{2m-4}}\) by simplifying the exponent:

  • First, we did any addition or subtraction within the exponent, in this case: \((3m-5) - (m-1) = 3m - 5 - m + 1\).
  • This calculation led to \(2m - 4\), which then became the single exponent in the root \(\sqrt{a^{2m-4}}\).
This process is crucial because it ensures that the expression is as simple as possible.
It eliminates potential errors and prepares the equation for further algebraic operations if necessary.

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

Simplify each expression by performing the indicated operation. $$ \frac{4+\sqrt{5}}{4-\sqrt{5}} $$

For the following problems, simplify the expressions. $$ \sqrt{y} \sqrt{y} $$

For the following problems, solve the equations. $$ \sqrt{3 x}=9 $$

For the following problems, solve the equations. $$ 2 \sqrt{a+2}-2=0 $$

Simplify each expression by performing the indicated operation. $$ (1+\sqrt{3})^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.