Chapter 12, Problem 25CCSR

To determine

Find the correct option.

Answer to Problem 25CCSR

  $C$

Explanation of Solution

Given information:

Simplify the expression,

  $\frac{27a^{\frac{5}{2}}{b}^{\frac{4}{3}}}{9a^{2}b}.$

Options:-

  $\begin{array}{l}A 3ab\\ B 18a^{\frac{1}{4}}{b}^{\frac{1}{6}}\\ C 3a^{\frac{1}{2}}{b}^{\frac{1}{3}}\\ D 27a^{\frac{1}{4}}{b}^{\frac{1}{6}}\end{array}$

Calculation:

Use the below written exponential law

  $\frac{a^m}{a^n}=a^{m-n},m,n\in \mathbb{Z}$

Consider the given expression:

  $\frac{27a^{\frac{5}{2}}{b}^{\frac{4}{3}}}{9a^{2}b}.$

Further solving

  $\frac{27}{9}{a}^{\frac{5}{2}}{b}^{\frac{4}{3}}{a}^{-2}{b}^{-1}=3a^{\frac{5}{2}}{b}^{\frac{4}{3}}{a}^{-2}{b}^{-1}$

Taking the like terms together

  $\begin{array}{l}3a^{\frac{5}{2}}{b}^{\frac{4}{3}}{a}^{-2}{b}^{-1}=3\left(a^{\frac{5}{2}}{a}^{-2}\right)\left(b^{\frac{4}{3}}{b}^{-1}\right)\\ =3\left(a^{\frac{5}{2}-2}\right)\left(b^{\frac{4}{3}+(-1)}\right)\\ =3\left(a^{\frac{5}{2}-2}\right)\left(b^{\frac{4}{3}-1}\right)\\ =3a^{\frac{1}{2}}{b}^{\frac{1}{3}}\end{array}$

Hence, the correct option is $3a^{\frac{1}{2}}{b}^{\frac{1}{3}}$

Hence, the correct option is $C$ .