Chapter 6.7, Problem 72HP

To determine

To Write :

An equation that can be solved by raising each side of the equation to the given power.

  $\begin{array}{l}a)\frac{3}{2}\text{ power }\\ \text{b)}\frac{5}{4}\text{ power}\\ \text{c)}\frac{7}{8}\text{ power}\end{array}$

Answer to Problem 72HP

  $\begin{array}{l}\text{a) }{x}^{\frac{2}{3}}-10=0\\ b)x^{\frac{4}{5}}-3=0\\ c)\text{ 3}{x}^{\frac{8}{7}}-9=0\end{array}$

Explanation of Solution

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 $a=\sqrt[n]{b}$
• Also, ${\left(\sqrt[n]{b}\right)}^n={\left(a\right)}^n=b$
• That is , ${\left(\sqrt[n]{b}\right)}^n={\left(b^{\frac{1}{n}}\right)}^n=b$
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$

Calculation:

  $\begin{array}{l}{x}^{\frac{2}{3}}-10=0\text{ can be solved by raising each side to the power }\frac{3}{2}\\ Solve:\\ x^{\frac{2}{3}}=\mathrm{10..........}[\text{add 10 each side]}\\ {\left(x^{\frac{2}{3}}\right)}^{\frac{3}{2}}={10}^{\frac{3}{2}}\\ x={10}^{\frac{3}{2}}\\ Similarly,\\ x^{\frac{4}{5}}-3=0\text{ can be solved by raising each side to the power }\frac{5}{4}\\ \text{ 3}{x}^{\frac{8}{7}}-9=0\text{ can be solved by raising each side to the power }\frac{7}{8}\\ \text{So, the equations can be:}\\ \text{a) }{x}^{\frac{2}{3}}-10=0\\ b)x^{\frac{4}{5}}-3=0\\ c)\text{ 3}{x}^{\frac{8}{7}}-9=0\end{array}$