livinglife100s8x

2022-04-25

Simplify this expression

$w{\left(\frac{Q}{1+\frac{w}{r}}\right)}^{2}+r{\left(\frac{Q}{1+\frac{r}{w}}\right)}^{2}$

Kendal Kelley

Beginner2022-04-26Added 16 answers

Observe that $\frac{a}{\frac{b}{c}}=\frac{ac}{b}$ but $\frac{\left(\frac{a}{b}\right)}{c}=\frac{a}{bc}$

Here,

$\frac{Q}{1+\frac{w}{r}}=\frac{Q}{\frac{r+w}{r}}=\frac{Qr}{r+w}$

$\Rightarrow w{\left(\frac{Q}{1+\frac{w}{r}}\right)}^{2}=\frac{{Q}^{2}{r}^{2}w}{{(r+w)}^{2}}$

Similarly,

$r{\left(\frac{Q}{1+\frac{r}{w}}\right)}^{2}=\frac{{Q}^{2}r{w}^{2}}{{(r+w)}^{2}}$

Adding we get,

$\frac{{Q}^{2}r{w}^{2}+{Q}^{2}{r}^{2}w}{{(r+w)}^{2}}=\frac{{Q}^{2}rw(r+w)}{{(r+w)}^{2}}=\frac{{Q}^{2}rw}{(r+w)}$

Now divide the numerator & the denominator by rw

Also, observe that the question itself has assumed that$rw(r+w)\ne 0$

Here,

Similarly,

Adding we get,

Now divide the numerator & the denominator by rw

Also, observe that the question itself has assumed that

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