Question

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Answer 1

Question :-

$\sf \Big(\dfrac{5^{-1} \times 7^2}{5^{-2} \times 7^{-4}}\Big)^\frac{7}{2} \times \Big(\dfrac{5^{-2} \times 7^3}{5^{3} \times 7^{-5}}\Big)^\frac{-5}{2}$

Answer :-

$\implies\sf \Big(\dfrac{5^{-1} \times 7^2}{5^{-2} \times 7^{-4}}\Big)^\frac{7}{2} \times \Big(\dfrac{5^{-2} \times 7^3}{5^{3} \times 7^{-5}}\Big)^\frac{-5}{2}$

$\implies\sf \Big(\dfrac{5^{-1}}{5^{-2}} \times \dfrac{7^2}{7^{-4}}\Big)^\frac{7}{2} \times \Big(\dfrac{5^{-2}}{5^{3}} \times \dfrac{7^3}{7^{-5}}\Big)^\frac{-5}{2}$

$\implies\sf [5^{-1-2} \times 7^{2-(-4)}]^\frac{7}{2} \times [5^{-2-3} \times 7^{3-(-5)}]^\frac{-5}{2}$

$\implies\sf (5^{-3} \times 7^{2 + 4})^\frac{7}{2} \times (5^{-5} \times 7^{3+5})^\frac{-5}{2}$

$\implies\sf (5^{-3} \times 7^{6})^\frac{7}{2} \times (5^{-5} \times 7^{8})^\frac{-5}{2}$

$\implies\sf (5^{-3})^\frac{7}{2} \times (7^{\cancel6})^\frac{7}{\cancel2} \times (5^{-5})^\frac{-5}{2}\times (7^{\cancel8})^\frac{-5}{\cancel2}$

$\implies\sf 5^\frac{-21}{2} \times 7^{3\times 7} \times 5^\frac{25}{2} \times 7^{-5 \times 4}$

$\implies\sf 5^{\frac{-21}{2} + \frac{25}{2}} \times 7^{21} \times 7^{-20}$

$\implies\sf 5^\frac{\cancel4}{\cancel2} \times 7^{21-20}$

$\implies\sf 5^2 \times 7^1$

$\implies\sf 25 \times 7$

$\implies\boxed{\sf 175}$

Answer 2

Given:

The following expression:

$(\frac{5^{-1} \times 7^{2}}{5^2\times7^{2}} )^{\frac72}\times( \frac{5^-2 \times 7^3}{5^3\times7^-5})^{\frac{-5}{2}$

To find:

The value of the expression

Solution:

$(\frac{5^{-1} \times 7^{2}}{5^2\times7^{2}} )^{\frac72}\times( \frac{5^-2 \times 7^3}{5^3\times7^-5})^{\frac{-5}{2}$

Taking all exponents of 5 and 7 to numerators

$(5^{-1-2} \times 7^{2+4})^{\frac72}\times( 5^{-2-3} \times 7^{3+5})^\frac{-5}{2}\\\\=( 5^{-3}\times 7^6)^\frac72 \times ( 5^{-5}\times 7^{8})^\frac{-5}{2}$

$= 5^{-3\times\frac72} \times 7^{6\times\frac72} \times 5^ {-5\times\frac{-5}{2} }\times 7^ { 8 \times \frac {-5}{2} } \\\\ = 5^ { \frac {-21}{2} +\frac {25}{2} } \times 7^ { 21-20}\\\\=5^2 \times 7\\\\=25 \times 7\\\\=175$

Answer:

175

Formulas used :

General rules and formulas of exponents

1. $1.\, a^x \times a^y =a^{x+y}\\\\2.\, \frac{ a^x \times b^p } {a^y \times b^q} = a^{x-y} \times b^{ \,p-q}\\\\3.\, ( a^x)^y=a^{xy}$