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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