Chapter 6.2, Problem 57WE

To determine

To Evaluate:

  $\sqrt[6]{{\left(xy\right)}^z}$ for x = 4 , y =3 , z =8

Answer to Problem 57WE

  $12\sqrt[6]{144}$

Explanation of Solution

Given Information:

Given expression to evaluate:

  $\sqrt[6]{{\left(xy\right)}^z}$

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 $\sqrt[4]{81}=\pm 3$ .
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$

    $a^{-n}=\frac{1}{a^n}$

• 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 : $\pm \sqrt[n]{a}$ ; 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 : $\sqrt[n]{0}=0$

Calculation:

• Given $\sqrt[6]{{\left(xy\right)}^z}$
• $\begin{array}{l}\sqrt[6]{{\left(xy\right)}^z}\\ S\text{ubstitute x = 4 , y = 3 , z = 8:}\\ =\sqrt[6]{{\left(4\cdot 3\right)}^8}\\ =\sqrt[6]{{12}^6\cdot {12}^2}\mathrm{.............}[a^{x+y}=a^x\cdot a^y]\\ =12\sqrt[6]{144}\end{array}$
• So, $12\sqrt[6]{144}$ is the solution.