# Evaluate ((sqrt10^1009))/(sqrt10^1011-sqrt10^1007)without using a calculator?

Aug 19, 2017

$\frac{{\sqrt{10}}^{1009}}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}} = \frac{10}{99} = 0. \overline{10}$

#### Explanation:

Note that $\sqrt{10} = {10}^{\frac{1}{2}}$ and if $a , b , c > 0$ then:

${a}^{b} \cdot {a}^{c} = {a}^{b + c} \text{ }$ and $\text{ } {\left({a}^{b}\right)}^{c} = {a}^{b c}$

So we find:

$\frac{{\sqrt{10}}^{1009}}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}} = {10}^{\frac{1009}{2}} / \left({10}^{\frac{1011}{2}} - {10}^{\frac{1007}{2}}\right)$

$\textcolor{w h i t e}{\frac{{\sqrt{10}}^{1009}}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}}} = {10}^{\frac{1007}{2} + 1} / \left({10}^{\frac{1007}{2} + 2} - {10}^{\frac{1007}{2} + 0}\right)$

$\textcolor{w h i t e}{\frac{{\sqrt{10}}^{1009}}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}}} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{{10}^{\frac{1007}{2}}}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{{10}^{\frac{1007}{2}}}}}} \cdot \left(\frac{10}{100 - 1}\right)$

$\textcolor{w h i t e}{\frac{{\sqrt{10}}^{1009}}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}}} = \frac{10}{99} = 0. \overline{10}$

Aug 19, 2017

#### Explanation:

${\sqrt{10}}^{1009} / \left({\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}\right)$

There is a common factor of ${\sqrt{10}}^{1007}$.

$= \frac{{\sqrt{10}}^{1007} \left({\sqrt{10}}^{2}\right)}{{\sqrt{10}}^{1007} \left({\sqrt{10}}^{4} - 1\right)}$

$= {\sqrt{10}}^{2} / \left({\sqrt{10}}^{4} - 1\right)$

$= \frac{10}{100 - 1} = \frac{10}{99} = 0. \overline{10}$