Solve {left(frac{9/10right)}^10}{{left(9/10right)}^10+10{left(9/10right)}^91/10}

$$\frac{ { \left( \frac{ 9 }{ 10 } \right) }^{ 10 } }{ { \left( \frac{ 9 }{ 10 } \right) }^{ 10 } +10 { \left( \frac{ 9 }{ 10 } \right) }^{ 9 } \frac{ 1 }{ 10 } }$$

Evaluate

$$\frac{9}{19}\approx 0.473684211$$

Solution Steps

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

$$\frac{\frac{3486784401}{10000000000}}{\left(\frac{9}{10}\right)^{10}+10\times \left(\frac{9}{10}\right)^{9}\times \frac{1}{10}}$$

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

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+10\times \left(\frac{9}{10}\right)^{9}\times \frac{1}{10}}$$

Calculate \frac{9}{10} to the power of 9 and get \frac{387420489}{1000000000}.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+10\times \frac{387420489}{1000000000}\times \frac{1}{10}}$$

Express 10\times \frac{387420489}{1000000000} as a single fraction.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+\frac{10\times 387420489}{1000000000}\times \frac{1}{10}}$$

Multiply 10 and 387420489 to get 3874204890.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+\frac{3874204890}{1000000000}\times \frac{1}{10}}$$

Reduce the fraction \frac{3874204890}{1000000000} to lowest terms by extracting and canceling out 10.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+\frac{387420489}{100000000}\times \frac{1}{10}}$$

Multiply \frac{387420489}{100000000} times \frac{1}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+\frac{387420489\times 1}{100000000\times 10}}$$

Do the multiplications in the fraction \frac{387420489\times 1}{100000000\times 10}.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+\frac{387420489}{1000000000}}$$

Least common multiple of 10000000000 and 1000000000 is 10000000000. Convert \frac{3486784401}{10000000000} and \frac{387420489}{1000000000} to fractions with denominator 10000000000.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401}{10000000000}+\frac{3874204890}{10000000000}}$$

Since \frac{3486784401}{10000000000} and \frac{3874204890}{10000000000} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3486784401}{10000000000}}{\frac{3486784401+3874204890}{10000000000}}$$

Add 3486784401 and 3874204890 to get 7360989291.

$$\frac{\frac{3486784401}{10000000000}}{\frac{7360989291}{10000000000}}$$

Divide \frac{3486784401}{10000000000} by \frac{7360989291}{10000000000} by multiplying \frac{3486784401}{10000000000} by the reciprocal of \frac{7360989291}{10000000000}.

$$\frac{3486784401}{10000000000}\times \frac{10000000000}{7360989291}$$

Multiply \frac{3486784401}{10000000000} times \frac{10000000000}{7360989291} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3486784401\times 10000000000}{10000000000\times 7360989291}$$

Cancel out 10000000000 in both numerator and denominator.

$$\frac{3486784401}{7360989291}$$

Reduce the fraction \frac{3486784401}{7360989291} to lowest terms by extracting and canceling out 387420489.

$$\frac{9}{19}$$

Factor

$$\frac{3 ^ {2}}{19} = 0.47368421052631576$$