Evaluate $\frac{({\sqrt{10}}^{1009})}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}}$ $((sqrt10^1009))/(sqrt10^1011-sqrt10^1007)$ without using a calculator?

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Evaluate $\frac{({\sqrt{10}}^{1009})}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}}$ $((sqrt10^1009))/(sqrt10^1011-sqrt10^1007)$ without using a calculator?

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Answer 1 · 2017-08-19T13:16:38

$\frac{{\sqrt{10}}^{1009}}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}} = \frac{10}{99} = 0 . ¯ ¯¯ ¯ 10$ $(sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007)) = 10/99 = 0.bar(10)$

Explanation:

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

$a^{b} \cdot a^{c} = a^{b + c}$ $a^b*a^c = a^(b+c)" "$ and ${(a^{b})}^{c} = a^{b c}$ $" "(a^b)^c = a^(bc)$

So we find:

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

$\frac{{\sqrt{10}}^{1009}}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}} = \frac{10^{\frac{1007}{2} + 1}}{10^{\frac{1007}{2} + 2} - 10^{\frac{1007}{2} + 0}}$ $color(white)((sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007))) = 10^(1007/2+1)/(10^(1007/2+2)-10^(1007/2+0))$

$color(white)((sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007))) = color(red)(cancel(color(black)(10^(1007/2))))/color(red)(cancel(color(black)(10^(1007/2))))*(10/(100-1))$

$color(white)((sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007))) = 10/99 = 0.bar(10)$

Answer 2 · 2017-08-19T16:07:12

Same answer, different notation.

Explanation:

$sqrt10^1009/(sqrt10^1011-sqrt10^1007)$

There is a common factor of $sqrt10^1007$ .

$= (sqrt10^1007(sqrt10^2))/(sqrt10^1007(sqrt10^4-1))$

$= sqrt10^2/(sqrt10^4-1)$

$= 10/(100-1) = 10/99 = 0.bar(10)$

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Evaluate $\frac{({\sqrt{10}}^{1009})}{{\sqrt{10}}^{1011} - {\sqrt{10}}^{1007}}$ $((sqrt10^1009))/(sqrt10^1011-sqrt10^1007)$ without using a calculator?

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