Chapter 7.5, Problem 1BCYP

To determine

To Simplify:

  $\sqrt[3]{27y^{12}{z}^7}$

Answer to Problem 1BCYP

  $3y^4{z}^2\sqrt[3]{z}$

Explanation of Solution

Given Information:

Given expression to simplify:

  $\sqrt[3]{27y^{12}{z}^7}$

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 : $\sqrt[n]{0}=0$

• 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: $\begin{array}{l}\text{For any real number a and b and any integer n > 1 , }\\ \sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b}\text{ , if n is even and a \& b are both nonnegative }\\ \text{or if n is odd}\text{.}\end{array}$

Calculation:

• Given $\sqrt[3]{27y^{12}{z}^7}$
• $\begin{array}{l}\sqrt[3]{27y^{12}{z}^7}\\ =\sqrt[3]{3^3\cdot {\left(y^4\right)}^3\cdot z^6\cdot z}\mathrm{.............................................}[a^{x+y}=a^x\cdot a^y]\\ =\sqrt[3]{3^3\cdot {\left(y^4\right)}^3\cdot {\left(z^2\right)}^3\cdot z}\mathrm{.........................................}[Factor\text{ into cubes]}\\ =\sqrt[3]{3^3}\cdot \sqrt[3]{{\left(y^4\right)}^3}\cdot \sqrt[3]{{\left(z^2\right)}^3}\cdot \sqrt[3]{z}\mathrm{..............................}\text{[Product Property Of Radicals]}\\ =3y^4{z}^2\sqrt[3]{z}\mathrm{..........................................................}[\text{Taking Absolute Value ]}\end{array}$
• Since , the index is 3 , which is odd , so there is only one root ; hence no need to use the absolute value symbol to find the principal root.
• So, $3y^4{z}^2\sqrt[3]{z}\text{ is the root}\text{.}$