i To make the powers 16 and 14 same We find the LCM of 6 4 is 12 and ii To make the powers 12 and 13 same LCM of 2 and 3 is 6

Selina Solutions Class 9 Mathematics Solutions for Exercise 1(B) in Chapter 1 - Chapter 1- Rational and Irrational Numbers

Question 15 Exercise 1(B)

Compare:

(i) $\sqrt[6]{15}$ and $\sqrt[4]{12}$

(ii) √𝟐𝟒 𝒂𝒏𝒅 $\sqrt[3]{35}$

Answer:

(i) $\sqrt[6]{15}=(15)^{\frac{1}{6}} \text { and } \sqrt[4]{12}=(12)^{\frac{1}{4}}$ To make the powers 1/6 and 1/4 same,

We find the L.C.M. of 6, 4 is 12

$\frac{1}{6} \times \frac{2}{2}=\frac{2}{12}$

and $\begin{array}{l} \frac{1}{4} \times \frac{3}{3}=\frac{3}{12} \\ \Rightarrow \sqrt[6]{15}=(15)^{\frac{1}{6}}=(15)^{\frac{2}{12}}=\left(15^{2}\right)^{\frac{1}{12}}=(225)^{\frac{1}{12}} \\ \text { and } \sqrt[4]{12}=(12)^{\frac{1}{4}}=(12)^{\frac{3}{12}}=\left(12^{3}\right)^{\frac{1}{12}}=(1728)^{\frac{1}{12}} \\ \Rightarrow 1272>225 \\ \Rightarrow(1728)^{\frac{1}{12}}>(225)^{\frac{1}{12}} \\ \Rightarrow \sqrt[4]{12}>\sqrt[6]{15} \end{array}$

(ii) $\sqrt{24}=(24)^{\frac{1}{2}} \text { and } \sqrt[3]{35}=(35)^{\frac{1}{3}}$

To make the powers 1/2 and 1/3 same,

L.C.M. of 2 and 3 is 6.

$\begin{array}{l} \frac{1}{2} \times \frac{3}{3}=\frac{3}{6}, \frac{1}{3} \times \frac{2}{2}=\frac{2}{6} \\ \Rightarrow(24)^{\frac{1}{2}}=(24)^{\frac{3}{6}}=\left(24^{3}\right)^{\frac{1}{6}}=(13824)^{\frac{1}{6}} \\ (35)^{\frac{1}{3}}=(35)^{\frac{2}{6}}=\left(35^{-2}\right)^{\frac{1}{6}}=(1225)^{\frac{1}{6}} \\ \Rightarrow 13824>1225 \\ \Rightarrow(13824)^{\frac{1}{6}}>\sqrt[3]{35} \\ \Rightarrow \sqrt{24}>\sqrt[3]{35} \end{array}$