Answer
$\frac{ ab}{ 5b+2a}$.
Work Step by Step
The given expression is
$=\frac{5a^{-1}-2b^{-1}}{25a^{-2}-4b^{-2}}$
Multiply the numerator and the denominator by $a^2b^2$.
$=\frac{a^2b^2}{a^2b^2}\cdot \frac{5a^{-1}-2b^{-1}}{25a^{-2}-4b^{-2}}$
Use the distributive property.
$=\frac{a^2b^2 \cdot 5a^{-1}-a^2b^2 \cdot 2b^{-1}}{a^2b^2 \cdot 25a^{-2}-a^2b^2 \cdot 4b^{-2}}$
Simplify.
$=\frac{ 5a^{-1+2}b^2- 2a^2b^{-1+2}}{ 25a^{-2+2}b^2- 4a^2b^{-2+2}}$
$=\frac{ 5a^{1}b^2- 2a^2b^{1}}{ 25a^{0}b^2- 4a^2b^{0}}$
$=\frac{ 5ab^2- 2a^2b}{ 25b^2- 4a^2}$
Factor the fraction.
Numerator $=5ab^2-2a^2b$.
Factor out the common terms.
$=ab(5b-2a)$
Denominator $=25b^2-4a^2$.
$=(5b)^2-(2a)^2$
Use the algebraic identity $a^2-b^2=(a+b)(a-b)$.
$=(5b+2a)(5b-2a)$.
Substitute all factors into the fraction.
$=\frac{ ab(5b-2a)}{ (5b+2a)(5b-2a)}$.
Cancel common terms.
$=\frac{ ab}{ 5b+2a}$.
