Chapter 6.5, Problem 2AGP

To determine

To Simplify:

  $\frac{\sqrt{a^9}}{\sqrt{b^5}}$

Answer to Problem 2AGP

  $\frac{a^4}{\left|b^3\right|}\sqrt{ab}$

Explanation of Solution

Given Information:

Given expression to simplify:

  $\frac{\sqrt{a^9}}{\sqrt{b^5}}$

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

  $\begin{array}{l}\text{For any real numbers a and b}\ne \text{0 and any integer}\\ \text{n >1,}\\ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},if\text{ all roots are defined}\text{.}\end{array}$

• Rationalizing the denominator :

To eliminate radicals from a denominator or fractions from a radicand , use this process.

To rationalize a denominator, multiply the numerator and denominator by a quantity so that the radicand has an exact root.

Calculation:

• Given$\frac{\sqrt{a^9}}{\sqrt{b^5}}$
• $\begin{array}{l}\frac{\sqrt{a^9}}{\sqrt{b^5}}\\ =\frac{\sqrt{a^9}}{\sqrt{b^5}}\cdot \frac{\sqrt{b}}{\sqrt{b}}\mathrm{.............}[\text{rationalize the denominator]}\\ \text{=}\frac{\sqrt{a^9}\cdot \sqrt{b}}{\sqrt{b^5}\cdot \sqrt{b}}\\ =\frac{\sqrt{a^9\cdot b}}{\sqrt{b^5\cdot b}}\mathrm{................}[\text{product property of radicals]}\\ \text{=}\frac{\sqrt{a^8\cdot a\cdot b}}{\sqrt{b^6}}\mathrm{............}[a^{x+y}=a^x\cdot a^y]\\ =\frac{\sqrt{{\left(a^4\right)}^2\cdot a\cdot b}}{\sqrt{{\left(b^3\right)}^2}}\mathrm{.......}[\text{factor into squares]}\\ =\frac{a^4\sqrt{ab}}{b^3}\\ =\frac{a^4}{b^3}\sqrt{ab}\end{array}$
• Since , the index is 2 , which is even and the solution has odd power, so we need to apply the absolute value symbol to find the principal root.
• So, $\frac{a^4}{\left|b^3\right|}\sqrt{ab}\text{ is the principal root}\text{.}$