Question

If the range of $x$ for which the expansion of $(4-7x{)}^{-\frac{2}{5}}$is valid is $(a,b)$ then value of $7(b-a)$ =

Answer 1

We know that, $(1\pm x{)}^n,n\in Q-W$ is valid iff $|x|<1$

so, $(4-7x{)}^{-\frac{2}{5}}={\left(4(1-\frac{7}{4}x)\right)}^{-\frac{2}{5}}=4^{-\frac{2}{5}}{(1-\frac{7x}{4})}^{\frac{-2}{5}}$

For the above expansion to be valid
$\left|\frac{7x}{4}\right|<1$
$\Rightarrow x\in (-\frac{4}{7},\frac{4}{7})$
$\therefore 7(b-a)=8$