"Simplify this expression w(Q1+wr)2+r(Q1+rw)2" solved and explained

livinglife100s8x

Answered question

2022-04-25

Simplify this expression

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

Kendal Kelley

Beginner2022-04-26Added 16 answers

Observe that

\frac{a}{\frac{b}{c}} = \frac{a c}{b}

but

\frac{(\frac{a}{b})}{c} = \frac{a}{b c}

Here,

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

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

Similarly,

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

Now divide the numerator & the denominator by rw
Also, observe that the question itself has assumed that

r w (r + w) \neq 0

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