Question

Simplify:
\[{\left( {\dfrac{{{5^{ - 1}} \times {7^2}}}{{{5^2} \times {7^{ - 4}}}}} \right)^{\dfrac{7}{2}}} \times {\left( {\dfrac{{{5^2} \times {7^3}}}{{{5^3} \times {7^{ - 5}}}}} \right)^{ - \dfrac{5}{2}}}\]

Answer 1

Hint – Whenever we come to such a type of problem always try to apply the properties which provide the required answer very easily. in this problem we will use the logarithmic properties $\left( {\dfrac{{{{\text{a}}^{\text{m}}}}}{{{{\text{a}}^{\text{n}}}}}{\text{ = }}{{\text{a}}^{{\text{m - n}}}},{{{\text{(}}{{\text{a}}^{\text{m}}}{\text{)}}}^{\text{n}}}{\text{ = }}{{\text{a}}^{{\text{mn}}}},{{\text{a}}^{{\text{ - m}}}}{\text{ = }}\dfrac{{\text{1}}}{{{{\text{a}}^{\text{m}}}}}} \right)$.

Complete step-by-step answer:
Given that
\[{\left( {\dfrac{{{5^{ - 1}} \times {7^2}}}{{{5^2} \times {7^{ - 4}}}}} \right)^{\dfrac{7}{2}}} \times {\left( {\dfrac{{{5^2} \times {7^3}}}{{{5^3} \times {7^{ - 5}}}}} \right)^{ - \dfrac{5}{2}}}\]
$ = {\left( {{5^{ - 1 - 2}} \times {7^{2 + 4}}} \right)^{\dfrac{7}{2}}} \times {\left( {{5^{ - 1}} \times {7^8}} \right)^{\dfrac{{ - 5}}{2}}}{\text{ using }}\dfrac{{{{\text{a}}^{\text{m}}}}}{{{{\text{a}}^{\text{n}}}}}{\text{ = }}{{\text{a}}^{{\text{m - n}}}}$
$ = {5^{\dfrac{{ - 21}}{2}}} \times {7^{21}} \times {5^{\dfrac{5}{2}}} \times {7^{ - 20}}{\text{ using (}}{{\text{a}}^{\text{m}}}{{\text{)}}^{\text{n}}}{\text{ = }}{{\text{a}}^{{\text{mn}}}}$
$ = {5^{\dfrac{{ - 21}}{2} + \dfrac{5}{2}}} \times {7^{21 - 20}}$
$ = {5^{\dfrac{{ - 16}}{2}}} \times {7^1}$
$ = {5^{ - 8}} \times 7$
$ = \dfrac{7}{{{5^8}}}{\text{ since }}{{\text{a}}^{{\text{ - m}}}}{\text{ = }}\dfrac{{\text{1}}}{{{{\text{a}}^{\text{m}}}}}$
$ = \dfrac{7}{{390625}}$

Note – Whenever you come to this type of problem, you must follow the exponents properties for getting the required answer in a simple and optimized way. As it is a very complicated problem please take care of the positive and negative powers.