Calculate: (2x^{-1}+3y^{-1})/(9x^2-4y^2)

Expression: $\frac{ 2{x}^{-1}+3{y}^{-1} }{ 9{x}^{2}-4{y}^{2} }$

Any expression raised to the power of $-1$ equals its reciprocal

$\frac{ 2 \times \frac{ 1 }{ x }+3{y}^{-1} }{ 9{x}^{2}-4{y}^{2} }$

Any expression raised to the power of $-1$ equals its reciprocal

$\frac{ 2 \times \frac{ 1 }{ x }+3 \times \frac{ 1 }{ y } }{ 9{x}^{2}-4{y}^{2} }$

Calculate the product

$\frac{ \frac{ 2 }{ x }+3 \times \frac{ 1 }{ y } }{ 9{x}^{2}-4{y}^{2} }$

Calculate the product

$\frac{ \frac{ 2 }{ x }+\frac{ 3 }{ y } }{ 9{x}^{2}-4{y}^{2} }$

Write all numerators above the least common denominator $xy$

$\frac{ \frac{ 2y+3x }{ xy } }{ 9{x}^{2}-4{y}^{2} }$

Simplify the complex fraction

$\frac{ 2y+3x }{ xy \times \left( 9{x}^{2}-4{y}^{2} \right) }$

Use ${a}^{2}-{b}^{2}=\left( a-b \right)\left( a+b \right)$ to factor the expression

$\frac{ 2y+3x }{ xy \times \left( 3x-2y \right) \times \left( 3x+2y \right) }$

Cancel out the common factor $3x+2y$

$\frac{ 1 }{ xy \times \left( 3x-2y \right) }$

Distribute $xy$ through the parentheses

$\frac{ 1 }{ 3{x}^{2}y-2x{y}^{2} }$