Solve:"

Given: $\frac{( 25)^{\frac{5}{2}} \ \times \ ( 729)^{\frac{1}{2}}}{( 125)^{\frac{5}{3}} \ \times \ ( 27)^{\frac{2}{3}} \ \times \ ( 8)^{\frac{4}{3}}}$

To find: Here we have to find the value of the given expression $\frac{( 25)^{\frac{5}{2}} \ \times \ ( 729)^{\frac{1}{2}}}{( 125)^{\frac{5}{3}} \ \times \ ( 27)^{\frac{2}{3}} \ \times \ ( 8)^{\frac{4}{3}}}$

Solution:

$\frac{( 25)^{\frac{5}{2}} \ \times \ ( 729)^{\frac{1}{2}}}{( 125)^{\frac{5}{3}} \ \times \ ( 27)^{\frac{2}{3}} \ \times \ ( 8)^{\frac{4}{3}}}$

$=\ \frac{\left( 5^{2}\right)^{\frac{5}{2}} \ \times \ \left( 27^{2}\right)^{\frac{1}{2}}}{\left( 5^{3}\right)^{\frac{5}{3}} \ \times \ \left( 3^{3}\right)^{\frac{2}{3}} \ \times \ \left( 2^{3}\right)^{\frac{4}{3}}}$

$=\ \frac{( 5)^{2\ \times \ \frac{5}{2}} \ \times \ ( 27)^{2\ \times \ \frac{1}{2}}}{( 5)^{3\ \times \ \frac{5}{3}} \ \times \ ( 3)^{3\ \times \ \frac{2}{3}} \ \times \ ( 2)^{3\ \times \ \frac{4}{3}}}$

$=\ \frac{( 5)^{5} \ \times \ ( 27)}{( 5)^{5} \ \times \ ( 3)^{2} \ \times \ ( 2)^{4}}$

$=\ \frac{( 3)^{3}}{( 3)^{2} \ \times \ ( 2)^{4}}$

$=\ \mathbf{\frac{3}{16}}$

So, the value of the given expression is $\frac{3}{16}$.

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Updated on: 10-Oct-2022

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