# Simplify:$\sqrt[lm]{\frac{x^{l}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n l]{\frac{x^{n}}{x^{l}}}$

Given:

$\sqrt[lm]{\frac{x^{l}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n l]{\frac{x^{n}}{x^{l}}}$

To do:

We have to simplify the given expression.

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

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

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

$\sqrt[lm]{\frac{x^{l}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n l]{\frac{x^{n}}{x^{l}}}=(x^{l-m})^{\frac{1}{l m}} \times (x^{m-n})^{\frac{1}{m n}} \times (x^{n-l})^{\frac{1}{n l}}$

$=x^{\frac{l-m}{l m}} \times x^{\frac{m-n}{m n}} \times x^{\frac{n-l}{n l}}$

$=x^{\frac{l-m}{l m}+\frac{m-n}{m n}+\frac{n-l}{n l}}$

$=x^{\frac{\ln -m n+l m-l n+m n-l m}{l m n}}$

$=x^{\frac{0}{l m n}}$

$=x^{0}$

$=1$

Hence, $\sqrt[lm]{\frac{x^{l}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n l]{\frac{x^{n}}{x^{l}}}=1$.

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Updated on: 10-Oct-2022

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