Assuming that $x, y, z$ are positive real numbers, simplify each of the following:$ \sqrt[5]{243 x^{10} y^{5} z^{10}} $
Given:
\( \sqrt[5]{243 x^{10} y^{5} z^{10}} \)
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,
$\sqrt[5]{243 x^{10} y^{5} z^{10}}=(243 x^{10} \times y^{5} \times z^{10})^{\frac{1}{5}}$
$=(3^{5} x^{10} \times y^{5} \times z^{10})^{\frac{1}{5}}$
$=3^{5 \times \frac{1}{5}} \times x^{10 \times \frac{1}{5}} \times y^{5 \times \frac{1}{5}} \times z^{10 \times \frac{1}{5}}$
$=3 x^{2} y z^{2}$
Hence, $\sqrt[5]{243 x^{10} y^{5} z^{10}}= 3 x^{2} y z^{2}$.
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