Chapter 6.7, Problem 71HP

To determine

To Explain :

The relationship between the index of the root of a variable in an equation and the power to which you raise each side of the equation to solve the equation.

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

Calculation:

  $\begin{array}{l}\text{The index of the root of a variable in an equation and the}\\ \text{power to which we need to raise each side of the equation }\\ \text{to solve the equation should be same}\text{.}\\ \text{Example: }\\ \sqrt[3]{x+20}=3\text{ is the equation}\\ \text{To solve it }\\ {\left(\sqrt[3]{x+20}\right)}^3=3^3\mathrm{............}[\text{eliminate the radical by raising to power 3=index]}\\ x+20=27\\ \text{and the equation can be solved easily in this manner}\text{.}\\ \\ \text{Hence, the index and the power to which we raise each side of the equation to solve }\\ \text{the equation are the same}\text{.}\end{array}$