Solve {left(frac{9/10right)}^10}{{left(9/10right)}^10+10{left(9/10right)}^99/10}

$$\frac{ { \left( \frac{ 9 }{ 10 } \right) }^{ 10 } }{ { \left( \frac{ 9 }{ 10 } \right) }^{ 10 } +10 { \left( \frac{ 9 }{ 10 } \right) }^{ 9 } \frac{ 9 }{ 10 } }$$

Evaluate

$$\frac{1}{11}\approx 0.090909091$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 9 and 1 to get 10.

$$\frac{\left(\frac{9}{10}\right)^{10}}{\left(\frac{9}{10}\right)^{10}+10\times \left(\frac{9}{10}\right)^{10}}$$

Calculate \frac{9}{10} to the power of 10 and get \frac{3486784401}{10000000000}.

$$\frac{\frac{3486784401}{10000000000}}{\left(\frac{9}{10}\right)^{10}+10\times \left(\frac{9}{10}\right)^{10}}$$

Calculate \frac{9}{10} to the power of 10 and get \frac{3486784401}{10000000000}.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+10\times \left(\frac{9}{10}\right)^{10}}$$

Calculate \frac{9}{10} to the power of 10 and get \frac{3486784401}{10000000000}.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+10\times \frac{3486784401}{10000000000}}$$

Express 10\times \frac{3486784401}{10000000000} as a single fraction.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+\frac{10\times 3486784401}{10000000000}}$$

Multiply 10 and 3486784401 to get 34867844010.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+\frac{34867844010}{10000000000}}$$

Since \frac{3486784401}{10000000000} and \frac{34867844010}{10000000000} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401+34867844010}{10000000000}}$$

Add 3486784401 and 34867844010 to get 38354628411.

$$\frac{\frac{3486784401}{10000000000}}{\frac{38354628411}{10000000000}}$$

Divide \frac{3486784401}{10000000000} by \frac{38354628411}{10000000000} by multiplying \frac{3486784401}{10000000000} by the reciprocal of \frac{38354628411}{10000000000}.

$$\frac{3486784401}{10000000000}\times \frac{10000000000}{38354628411}$$

Multiply \frac{3486784401}{10000000000} times \frac{10000000000}{38354628411} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3486784401\times 10000000000}{10000000000\times 38354628411}$$

Cancel out 10000000000 in both numerator and denominator.

$$\frac{3486784401}{38354628411}$$

Reduce the fraction \frac{3486784401}{38354628411} to lowest terms by extracting and canceling out 3486784401.

$$\frac{1}{11}$$

Factor

$$\frac{1}{11} = 0.09090909090909091$$