Chapter 7.5, Problem 38PPS

To determine

To Simplify:

  $(8\sqrt{5}-6\sqrt{3})(8\sqrt{5}+6\sqrt{3})$

Answer to Problem 38PPS

212

Explanation of Solution

Given Information:

Given expression to simplify:

  $(8\sqrt{5}-6\sqrt{3})(8\sqrt{5}+6\sqrt{3})$

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 $a=\sqrt[n]{b}$
• Example : Because ${(-3)}^4=81$ , -3 is a fourth root of 81 and 3 is a principal root. That is $\sqrt[4]{81}=\pm 3$ .
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$
• Product Property of Radicals: $\begin{array}{l}\text{For any real number a and b and any integer n > 1 , }\\ \sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b}\text{ , if n is even and a \& b are both nonnegative }\\ \text{or if n is odd}\text{.}\end{array}$
• Like radical expressions : If the radicals have both the index and radicand identical , then they are like radicals.
• In order to add or subtract , the radicals must be like radicals.

Calculation:

• Given $(8\sqrt{5}-6\sqrt{3})(8\sqrt{5}+6\sqrt{3})$
• $\begin{array}{l}(8\sqrt{5}-6\sqrt{3})(8\sqrt{5}+6\sqrt{3})\\ $ ={(8\sqrt{5})}^2-{(6\sqrt{3})}^2\mathrm{....................}[(a+b)(a-b)=a^2-b^2]$=64(5)-36(3)\\ =320-108\\ =212\end{array}$
• So, the answer is 212.