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$\fraction{9 y v}{\exponential{y}{2} - \exponential{v}{2}} + \fraction{y - v}{y + v} $
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\frac{9yv}{\left(y+v\right)\left(y-v\right)}+\frac{y-v}{y+v}
Factor y^{2}-v^{2}.
\frac{9yv}{\left(y+v\right)\left(y-v\right)}+\frac{\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y+v\right)\left(y-v\right) and y+v is \left(y+v\right)\left(y-v\right). Multiply \frac{y-v}{y+v} times \frac{y-v}{y-v}.
\frac{9yv+\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)}
Since \frac{9yv}{\left(y+v\right)\left(y-v\right)} and \frac{\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)} have the same denominator, add them by adding their numerators.
\frac{9yv+y^{2}-yv-yv+v^{2}}{\left(y+v\right)\left(y-v\right)}
Do the multiplications in 9yv+\left(y-v\right)\left(y-v\right).
\frac{7yv+y^{2}+v^{2}}{\left(y+v\right)\left(y-v\right)}
Combine like terms in 9yv+y^{2}-yv-yv+v^{2}.
\frac{7yv+y^{2}+v^{2}}{y^{2}-v^{2}}
Expand \left(y+v\right)\left(y-v\right).
\frac{9yv}{\left(y+v\right)\left(y-v\right)}+\frac{y-v}{y+v}
Factor y^{2}-v^{2}.
\frac{9yv}{\left(y+v\right)\left(y-v\right)}+\frac{\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y+v\right)\left(y-v\right) and y+v is \left(y+v\right)\left(y-v\right). Multiply \frac{y-v}{y+v} times \frac{y-v}{y-v}.
\frac{9yv+\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)}
Since \frac{9yv}{\left(y+v\right)\left(y-v\right)} and \frac{\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)} have the same denominator, add them by adding their numerators.
\frac{9yv+y^{2}-yv-yv+v^{2}}{\left(y+v\right)\left(y-v\right)}
Do the multiplications in 9yv+\left(y-v\right)\left(y-v\right).
\frac{7yv+y^{2}+v^{2}}{\left(y+v\right)\left(y-v\right)}
Combine like terms in 9yv+y^{2}-yv-yv+v^{2}.
\frac{7yv+y^{2}+v^{2}}{y^{2}-v^{2}}
Expand \left(y+v\right)\left(y-v\right).