Problem 20 Simplify each expression as comp... [FREE SOLUTION]

91影视

Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 10: Problem 20

Simplify each expression as completely as possible. Be sure your answers are in simplest radical form. Assume that all variables appearing under radical signs are nonnegative. $$\sqrt{24 a^{12}}$$

Short Answer

Expert verified

2a^6 \sqrt{6}

Step by step solution

- Factor the Radicand

First, factor the number and variable inside the square root. Factoring 24 gives us: \[ 24 = 2^3 \times 3 \] The variable part is already a perfect power: \[ a^{12} \]

- Apply the Square Root to Each Factor

Apply the square root to the constants and the variable part separately: \[ \ \sqrt{24 a^{12}} = \sqrt{2^3 \times 3 \times a^{12}} \]

- Simplify the Expressions Inside the Square Root

Simplify the expressions inside the square root by pairing the prime factors and using the properties of square roots: \[ \sqrt{2^3 \times 3 \times a^{12}} = \sqrt{2^2 \times 2 \times 3 \times a^{12}} = \sqrt{(2^2) \times 2 \times 3 \times a^{12}} \]

- Extract the Squares from the Square Root

Extract the squares from the square root: \[ \sqrt{(2^2) \times 2 \times 3 \times a^{12}} = 2 \times \sqrt{2 \times 3 \times a^{12}} = 2 \times \sqrt{6 \times a^{12}} \]

- Simplify the Variable Part

Since $a^{12}$ is a perfect square: \[ a^{12} = (a^6)^2 \] We get: \[ \sqrt{a^{12}} = a^6 \]

- Combine the Results

Combine the simplified constants and variable parts: \[ 2 \times a^6 \sqrt{6} \]

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

A square root is a value that, when multiplied by itself, gives the original number. We denote this as $ \sqrt{x} $. For example, since $ 3 \times 3 = 9 \, \sqrt{9} = 3 $. Working with square roots involves both recognizing and simplifying these roots.
In math, a square root helps us understand the relationship between numbers better. It's important to simplify square roots to make calculations easier and more understandable. For instance, $ \sqrt{24} $ can be broken down and simplified to make the math straightforward.

Factoring Radicands

Understanding the properties of square root simplifies computations when dealing with radicals. Some key properties include:

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Simplify each expression as completely as possible. Be sure your answers are in simplest radical form. Assume that all variables appearing under radical signs are nonnegative. $$\sqrt{24 a^{12}}$$

Short Answer

Step by step solution

- Factor the Radicand

- Apply the Square Root to Each Factor

- Simplify the Expressions Inside the Square Root

- Extract the Squares from the Square Root

- Simplify the Variable Part

- Combine the Results

Key Concepts

Square Roots

Factoring Radicands

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Decision Maths

Probability and Statistics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.