Simplify:(9)^1/3 × (27)^1/2 / (3)^1/6 × (3)^-2/3 

Solution:

Given, the expression is $\frac{(9)^{\frac{1}{3}}\times (27)^{\frac{1}{2}}}{(3)^{\frac{1}{6}}\times (3)^{\frac{-2}{3}}}$

We have to simplify the expression.

$(9)^{\frac{1}{3}}=(3^{2})^{\frac{1}{3}}=(3)^{\frac{2}{3}}$

$(27)^{\frac{-1}{2}}=(3^{3})^{\frac{-1}{2}}=(3)^{\frac{-3}{2}}$

We know $(a^{m})^{n}=a^{mn}$

$(9)^{\frac{1}{3}}\times (27)^{\frac{-1}{2}}=(3)^{\frac{2}{3}}\times (3)^{\frac{-3}{2}}$

We know $a^{m}\times a^{n}=a^{m+n}$

$(3)^{\frac{2}{3}}\times (3)^{\frac{-3}{2}}=(3)^{\frac{2}{3}+\frac{-3}{2}}=(3)^{\frac{-5}{6}}$

$(3)^{\frac{1}{6}}\times (3)^{\frac{-2}{3}}=(3)^{\frac{1}{6}+\frac{-2}{3}}=(3)^{\frac{-3}{6}}$

Now, $\frac{(9)^{\frac{1}{3}}\times (27)^{\frac{1}{2}}}{(3)^{\frac{1}{6}}\times (3)^{\frac{-2}{3}}}$ = $\frac{(3)^{\frac{-5}{6}}}{(3)^{\frac{-3}{6}}}$

We know $\frac{a^{m}}{a^{n}}=a^{m-n}$

= $(3)^{\frac{-5}{6}-\frac{-3}{6}}$

= $(3)^{\frac{-5}{6}+\frac{3}{6}}$

= $(3)^{\frac{-5+3}{6}}$

= $(3)^{\frac{-2}{6}}$

= $(3)^{\frac{-1}{3}}$

Therefore, the simplified expression is $(3)^{\frac{-1}{3}}$

✦ Try This: simplify : $\frac{(27)^{\frac{1}{3}}\times (9)^{\frac{1}{2}}}{(216)^{\frac{1}{6}}\times (3)^{\frac{-4}{3}}}$

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1

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NCERT Exemplar Class 9 Maths Exercise 1.3 Problem 14(v)

Simplify:(9)^1/3 × (27)^1/2 / (3)^1/6 × (3)^-2/3 

Summary:

If r is a rational number and s is an irrational number, then r+s and r-s are irrationals. The simplified form of the expression is 3^-⅓

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