Question
Compare: $\sqrt{24}$ and $\sqrt[3]{35}$
✓
Answer
$\sqrt{24}=(24)^{\frac{1}{2}} \text { and } \sqrt[3]{35}=35^{\frac{1}{3}}$
$\text{L.C.M}$. of $2$ and $3$ is $6 .$
$\frac{1}{2} \times \frac{3}{3}=\frac{3}{6}, \frac{1}{3} \times \frac{2}{2}=\frac{2}{6}$
$ \Rightarrow 24^{\frac{1}{2}}=24^{\frac{3}{6}}=\left(24^3\right)^{\frac{1}{6}}=(13824)^{\frac{1}{6}}$
$ \Rightarrow 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}$
$\text{L.C.M}$. of $2$ and $3$ is $6 .$
$\frac{1}{2} \times \frac{3}{3}=\frac{3}{6}, \frac{1}{3} \times \frac{2}{2}=\frac{2}{6}$
$ \Rightarrow 24^{\frac{1}{2}}=24^{\frac{3}{6}}=\left(24^3\right)^{\frac{1}{6}}=(13824)^{\frac{1}{6}}$
$ \Rightarrow 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}$
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