Answer 144 | a | b 2 2 is the principal root . Explanation Given Information: Given expression to simplify: 6 3 a b ⋅ 4 24 a b 3 Concept Used : Definition of nth root: Words : For any real numbers a and b , and any positive integer n , if a n = b then a is the nth root of b . That is a = b n Example : Because ( − 3 ) 4 = 81 , -3 is a fourth root of 81 and 3 is a principal root. That is 81 4 = ± 3 . b n H e r e b = r a d i c a n d n = i n d e x = r a d i c a l s i g n The non-negative root is called the Principal Root. Suppose n is an integer greater than 1, and a is a real number. a n is even n is odd a > 0 1 unique positive and 1 unique negative real root : ± a n ; positive root is principal root. 1 unique positive and 0 negative real root : a n a < 0 0 real roots 0 positive and 1 negative real root : − a n a=0 1 real root : 0 n = 0 1 real root : 0 n = 0 Pure Imaginary Numbers : Square roots of negative real numbers: for any positive real number b , − b 2 = b 2 − 1 , or bi where − 1 = i i = i o t a When you find an even root of an even power and the result is an odd power , you must use the absolute value of the result to ensure that the answer is nonnegative. If the index is odd, there is only one real root. Therefore, there is no principal root when index is odd , and absolute value symbol are not needed. Product Property of Radicals: For any real number a and b and any integer n > 1 , a b n = a n ⋅ b n , if n is even and a & b are both nonnegative or if n is odd . Quotient Property of Radicals: For any real numbers a and b ≠ 0 and any integer n >1, a b n = a n b n , i f all roots are defined . Rationalizing the denominator : To eliminate radicals from a denominator or fractions from a radicand , use this process. To rationalize a denominator, multiply the numerator and denominator by a quantity so that the radicand has an exact root. Calculation: Given 6 3 a b ⋅ 4 24 a b 3 6 3 a b ⋅ 4 24 a b 3 = 24 3 a b ⋅ 24 a b 3 .............. [ Product property of radicals] = 24 6 2 ⋅ 2 ⋅ a 2 ( b 2 ) 2 .......... [ Factoring into squares] = 24 ⋅ 6 ⋅ a ⋅ b 2 2 = 144 a b 2 2 Since , the index is 2 , which is even and the result has odd exponents , so we need to use the absolute value symbol to find the principal root. So, 144 | a | b 2 2 is the principal root . − 144 | a | b 2 2 is the other root .

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ISBN: 9780076639908

Author: McGraw-Hill Glencoe

Publisher: MCG

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Question

Chapter 6.5, Problem 28PPS

To determine

To Simplify:

$63ab⋅424ab3$

Expert Solution & Answer

Answer to Problem 28PPS

$144|a|b22 is the principal root.$

Explanation of Solution

Given Information:

Given expression to simplify:

$63ab⋅424ab3$

Concept Used :

Definition of nth root:

Words : For any real numbers a and b , and any positive integer n , if $an=b$ then a is the nth root of b . That is $a=bn$
Example : Because $(−3)4=81$ , -3 is a fourth root of 81 and 3 is a principal root. That is $814=±3$ .
$bnHereb=radicandn=index=radicalsign$

The non-negative root is called the Principal Root.
Suppose n is an integer greater than 1, and a is a real number.

a	n is even	n is odd
a > 0	1 unique positive and 1 unique negative real root : $±an$ ; positive root is principal root.	1 unique positive and 0 negative real root : $an$
a < 0	0 real roots	0 positive and 1 negative real root : $−an$
a=0	1 real root : $0n=0$	1 real root : $0n=0$

Pure Imaginary Numbers :

Square roots of negative real numbers: for any positive real number b , $−b2=b2−1$ , or bi where

$−1=ii=iota$

When you find an even root of an even power and the result is an odd power , you must use the absolute value of the result to ensure that the answer is nonnegative.
If the index is odd, there is only one real root. Therefore, there is no principal root when index is odd , and absolute value symbol are not needed.
Product Property of Radicals: $For any real number a and b and any integer n > 1 , abn=an⋅bn , if n is even and a & b are both nonnegative or if n is odd.$
Quotient Property of Radicals:

$For any real numbers a and b≠0 and any integern >1,abn=anbn,if all roots are defined.$

Rationalizing the denominator :

To eliminate radicals from a denominator or fractions from a radicand , use this process.

To rationalize a denominator, multiply the numerator and denominator by a quantity so that the radicand has an exact root.

Calculation:

Given $63ab⋅424ab3$
$63ab⋅424ab3=243ab⋅24ab3..............[Product property of radicals]=2462⋅2⋅a2(b2)2..........[Factoring into squares]=24⋅6⋅a⋅b22=144ab22$
Since , the index is 2 , which is even and the result has odd exponents , so we need to use the absolute value symbol to find the principal root.
So, $144|a|b22 is the principal root .−144|a|b22 is the other root.$

Chapter 6.5, Problem 27PPS

Chapter 6.5, Problem 29PPS

Chapter 6 Solutions

Glencoe Algebra 2 Student Edition C2014

Ch. 6.1 - Prob. 1AGP Ch. 6.1 - Prob. 1BGP Ch. 6.1 - Prob. 2AGP Ch. 6.1 - Prob. 2BGP Ch. 6.1 - Prob. 3AGP Ch. 6.1 - Prob. 3BGP Ch. 6.1 - Prob. 4GP Ch. 6.1 - Prob. 1CYU Ch. 6.1 - Prob. 2CYU Ch. 6.1 - Prob. 3CYU

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6 3 a b ⋅ 4 24 a b 3