/*! 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 46 Simplify. $$ \sqrt{81 p^{24}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 46

Simplify. $$ \sqrt{81 p^{24} q^{6}} $$

Short Answer

Expert verified
The simplified form is \ 9 p^{12} q^3 \.

Step by step solution

01

- Simplify the Constant

First, simplify the constant inside the square root. The square root of 81 is 9 because \(9^2 = 81\). Thus, \[ \sqrt{81 p^{24} q^{6}} = \sqrt{81} \cdot \sqrt{p^{24}} \cdot \sqrt{q^{6}} \Rightarrow 9 \cdot \sqrt{p^{24}} \cdot \sqrt{q^{6}} \].
02

- Simplify the Variable with Power of 24

Next, simplify \sqrt{p^{24}}. The square root of \p^{24} is \p^{24/2}. This simplifies to \p^{12}. Thus, \[ 9 \cdot p^{12} \cdot \sqrt{q^{6}} \].
03

- Simplify the Variable with Power of 6

Now, simplify \sqrt{q^6}. The square root of \q^6 is \q^{6/2}. This simplifies to \q^3. Thus, \[ 9 \cdot p^{12} \cdot q^3 \Rightarrow 9 p^{12} q^3 \].
04

- Combine All Simplified Parts

Combine all the simplified parts: \[ 9 \cdot p^{12} \cdot q^3 = 9 p^{12} q^3 \]. Therefore, \sqrt{81 p^{24} q^{6}}\ is equal to \9 p^{12} q^3\.

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.

square roots
Square roots are a fundamental concept in algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 81 is 9 since \(9 \times 9 = 81\). Square roots are denoted by the symbol \(\backslashsqrt{}\). When simplifying expressions involving square roots, it's helpful to recognize perfect squares鈥攏umbers like 1, 4, 9, 16, 25, etc. Understanding how to separate and simplify parts of an expression under a square root can make the process clearer; in the given exercise, we handled the constant and the variables separately to keep things neat.

Here's a quick process to remember:
  • Identify and simplify the constant inside the square root.
  • Break down variables under the square root into simpler components.
  • Apply the property \(\sqrt{a\cdot b} = \sqrt{a} \cdot \sqrt{b}\) to simplify things further.
exponents
Exponents represent the number of times a number, known as the base, is multiplied by itself. For example, \(p^{24}\) means that p is multiplied by itself 24 times. When dealing with square roots and exponents together, we can use the fractional exponent notation to simplify the calculations.

In simpler steps:
  • Understand that \(\sqrt{x^n} = x^{n/2}\).
  • Apply this rule to each term involving exponents.
  • Simplify the exponents by dividing by 2.

For instance, in the original exercise:
  • \texponent of \(p^{24}\) simplifies to \(p^{12}\) since \(24\div 2 = 12\).
  • \texponent of \(q^{6}\) simplifies to \(q^{3}\) since \(6\div 2 = 3\).
algebraic simplification
Algebraic simplification is the process of making expressions easier to work with by reducing them to their simplest form. The given problem showcases several relevant techniques:

First, use properties of square roots and exponents to simplify each component separately.
  • Handle constants and variables independently.
  • Use the rule \(\sqrt{a\cdot b} = \sqrt{a} \cdot \sqrt{b}\).
In the final step, combine all simplified parts to get the simplest form. In short:
  • Break down the expression.
  • Simplify each element.
  • Reassemble the simplified parts.
Algebraic simplification helps in reducing complex expressions and allows you to solve equations and inequalities more effectively.

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

In the following exercises, solve. \(\sqrt{2 m-3}-5=0\)

In the following exercises, solve. \(\sqrt{4 u+2}-6=0\)

In the following exercises, solve. (a) \(\sqrt{s-8}-s=-8\) (b) \(\sqrt{w+25}-w+5=0\)

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{s}-\sqrt{6}}{\sqrt{s}+\sqrt{6}} $$

In the following exercises, solve. \(\sqrt{2 n-1}-3=0\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision 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.