Solve 0.001/1-0.999^99+0.001/10+0.999^9*0.05=

$$\frac{ \frac{ 0.001 }{ 1- { 0.999 }^{ 9 } } 9+0.001 }{ 10 } + { 0.999 }^{ 9 } \times 0.05=$$

Evaluate

$$\frac{26901657242171803576371557275805444564940815847017999}{179281677482518320719820020000000000000000000000000000}\approx 0.150052463$$

Solution Steps

Calculate 0.999 to the power of 9 and get 0.991035916125874083964008999.

$$\frac{\frac{0.001}{1-0.991035916125874083964008999}\times 9+0.001}{10}+0.999^{9}\times 0.05$$

Subtract 0.991035916125874083964008999 from 1 to get 0.008964083874125916035991001.

$$\frac{\frac{0.001}{0.008964083874125916035991001}\times 9+0.001}{10}+0.999^{9}\times 0.05$$

Expand \frac{0.001}{0.008964083874125916035991001} by multiplying both numerator and the denominator by 1000000000000000000000000000.

$$\frac{\frac{1000000000000000000000000}{8964083874125916035991001}\times 9+0.001}{10}+0.999^{9}\times 0.05$$

Express \frac{1000000000000000000000000}{8964083874125916035991001}\times 9 as a single fraction.

$$\frac{\frac{1000000000000000000000000\times 9}{8964083874125916035991001}+0.001}{10}+0.999^{9}\times 0.05$$

Multiply 1000000000000000000000000 and 9 to get 9000000000000000000000000.

$$\frac{\frac{9000000000000000000000000}{8964083874125916035991001}+0.001}{10}+0.999^{9}\times 0.05$$

Convert decimal number 0.001 to fraction \frac{1}{1000}.

$$\frac{\frac{9000000000000000000000000}{8964083874125916035991001}+\frac{1}{1000}}{10}+0.999^{9}\times 0.05$$

Least common multiple of 8964083874125916035991001 and 1000 is 8964083874125916035991001000. Convert \frac{9000000000000000000000000}{8964083874125916035991001} and \frac{1}{1000} to fractions with denominator 8964083874125916035991001000.

$$\frac{\frac{9000000000000000000000000000}{8964083874125916035991001000}+\frac{8964083874125916035991001}{8964083874125916035991001000}}{10}+0.999^{9}\times 0.05$$

Since \frac{9000000000000000000000000000}{8964083874125916035991001000} and \frac{8964083874125916035991001}{8964083874125916035991001000} have the same denominator, add them by adding their numerators.

$$\frac{\frac{9000000000000000000000000000+8964083874125916035991001}{8964083874125916035991001000}}{10}+0.999^{9}\times 0.05$$

Add 9000000000000000000000000000 and 8964083874125916035991001 to get 9008964083874125916035991001.

$$\frac{\frac{9008964083874125916035991001}{8964083874125916035991001000}}{10}+0.999^{9}\times 0.05$$

Express \frac{\frac{9008964083874125916035991001}{8964083874125916035991001000}}{10} as a single fraction.

$$\frac{9008964083874125916035991001}{8964083874125916035991001000\times 10}+0.999^{9}\times 0.05$$

Multiply 8964083874125916035991001000 and 10 to get 89640838741259160359910010000.

$$\frac{9008964083874125916035991001}{89640838741259160359910010000}+0.999^{9}\times 0.05$$

Calculate 0.999 to the power of 9 and get 0.991035916125874083964008999.

$$\frac{9008964083874125916035991001}{89640838741259160359910010000}+0.991035916125874083964008999\times 0.05$$

Multiply 0.991035916125874083964008999 and 0.05 to get 0.04955179580629370419820044995.

$$\frac{9008964083874125916035991001}{89640838741259160359910010000}+0.04955179580629370419820044995$$

Convert decimal number 0.04955179580629370419820044995 to fraction \frac{991035916125874083964008999}{10000000000}. Reduce the fraction \frac{991035916125874083964008999}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{9008964083874125916035991001}{89640838741259160359910010000}+\frac{991035916125874083964008999}{20000000000000000000000000000}$$

Least common multiple of 89640838741259160359910010000 and 20000000000000000000000000000 is 179281677482518320719820020000000000000000000000000000. Convert \frac{9008964083874125916035991001}{89640838741259160359910010000} and \frac{991035916125874083964008999}{20000000000000000000000000000} to fractions with denominator 179281677482518320719820020000000000000000000000000000.

$$\frac{18017928167748251832071982002000000000000000000000000}{179281677482518320719820020000000000000000000000000000}+\frac{8883729074423551744299575273805444564940815847017999}{179281677482518320719820020000000000000000000000000000}$$

Since \frac{18017928167748251832071982002000000000000000000000000}{179281677482518320719820020000000000000000000000000000} and \frac{8883729074423551744299575273805444564940815847017999}{179281677482518320719820020000000000000000000000000000} have the same denominator, add them by adding their numerators.

$$\frac{18017928167748251832071982002000000000000000000000000+8883729074423551744299575273805444564940815847017999}{179281677482518320719820020000000000000000000000000000}$$

Add 18017928167748251832071982002000000000000000000000000 and 8883729074423551744299575273805444564940815847017999 to get 26901657242171803576371557275805444564940815847017999.

$$\frac{26901657242171803576371557275805444564940815847017999}{179281677482518320719820020000000000000000000000000000}$$