Chapter 7, Problem 47SGR

To determine

To Simplify:

  $\sqrt[4]{a^8{b}^{12}}$

Answer to Problem 47SGR

  $a^2\left|b^3\right|$

Explanation of Solution

Given Information:

Given expression to simplify:

  $\sqrt[4]{a^8{b}^{12}}$

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 $a=\sqrt[n]{b}$
• Example : Because ${(-3)}^4=81$ , -3 is a fourth root of 81 and 3 is a principal root. That is $\sqrt[4]{81}=\pm 3$ .
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$
• 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 : $\pm \sqrt[n]{a}$ ; positive root is principal root.	1 unique positive and 0 negative real root : $\sqrt[n]{a}$
a < 0	0 real roots	0 positive and 1 negative real root : $-\sqrt[n]{a}$
a=0	1 real root : $\sqrt[n]{0}=0$	1 real root : $\sqrt[n]{0}=0$

• Pure Imaginary Numbers :
• Square roots of negative real numbers: for any positive real number b, $\sqrt{-b^2}=\sqrt{b^2}\sqrt{-1}$ , or $$ bi where

  $\begin{array}{l}\sqrt{-1}=i\\ i=iota\end{array}$

• 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.

Calculation:

• Given $\sqrt[4]{a^8{b}^{12}}$

    $\begin{array}{l}\sqrt[4]{a^8{b}^{12}}=\sqrt[4]{{\left(a^2{b}^3\right)}^4}=a^2{b}^3\\ \text{Since the index is 4, which is odd and the power of the variable 'b' in the }\\ \text{solution is 3 , which is odd, we need to apply the absolute value symbol}\text{.}\\ \text{So, the principal root is }{a}^2\left|b^3\right|\text{ and the other root is }-a^2\left|b^3\right|\end{array}$

  So, $\sqrt[4]{a^8{b}^{12}}=a^2\left|b^3\right|$