Solve 8/1.0707+8/1.0707^2+8/1.0707^3+8+102/1.0707^4

$$\frac{ 8 }{ 1.0707 } + \frac{ 8 }{ { 1.0707 }^{ 2 } } + \frac{ 8 }{ { 1.0707 }^{ 3 } } + \frac{ 8+102 }{ 1.070 { 7 }^{ 4 } }$$

Evaluate

$$\frac{6625462911659113000}{315340205109169401}\approx 21.010523886$$

Solution Steps

Expand \frac{8}{1.0707} by multiplying both numerator and the denominator by 10000.

$$\frac{80000}{10707}+\frac{8}{1.0707^{2}}+\frac{8}{1.0707^{3}}+\frac{8+102}{1.07\times 7^{4}}$$

Calculate 1.0707 to the power of 2 and get 1.14639849.

$$\frac{80000}{10707}+\frac{8}{1.14639849}+\frac{8}{1.0707^{3}}+\frac{8+102}{1.07\times 7^{4}}$$

Expand \frac{8}{1.14639849} by multiplying both numerator and the denominator by 100000000.

$$\frac{80000}{10707}+\frac{800000000}{114639849}+\frac{8}{1.0707^{3}}+\frac{8+102}{1.07\times 7^{4}}$$

Least common multiple of 10707 and 114639849 is 114639849. Convert \frac{80000}{10707} and \frac{800000000}{114639849} to fractions with denominator 114639849.

$$\frac{856560000}{114639849}+\frac{800000000}{114639849}+\frac{8}{1.0707^{3}}+\frac{8+102}{1.07\times 7^{4}}$$

Since \frac{856560000}{114639849} and \frac{800000000}{114639849} have the same denominator, add them by adding their numerators.

$$\frac{856560000+800000000}{114639849}+\frac{8}{1.0707^{3}}+\frac{8+102}{1.07\times 7^{4}}$$

Add 856560000 and 800000000 to get 1656560000.

$$\frac{1656560000}{114639849}+\frac{8}{1.0707^{3}}+\frac{8+102}{1.07\times 7^{4}}$$

Calculate 1.0707 to the power of 3 and get 1.227448863243.

$$\frac{1656560000}{114639849}+\frac{8}{1.227448863243}+\frac{8+102}{1.07\times 7^{4}}$$

Expand \frac{8}{1.227448863243} by multiplying both numerator and the denominator by 1000000000000.

$$\frac{1656560000}{114639849}+\frac{8000000000000}{1227448863243}+\frac{8+102}{1.07\times 7^{4}}$$

Least common multiple of 114639849 and 1227448863243 is 1227448863243. Convert \frac{1656560000}{114639849} and \frac{8000000000000}{1227448863243} to fractions with denominator 1227448863243.

$$\frac{17736787920000}{1227448863243}+\frac{8000000000000}{1227448863243}+\frac{8+102}{1.07\times 7^{4}}$$

Since \frac{17736787920000}{1227448863243} and \frac{8000000000000}{1227448863243} have the same denominator, add them by adding their numerators.

$$\frac{17736787920000+8000000000000}{1227448863243}+\frac{8+102}{1.07\times 7^{4}}$$

Add 17736787920000 and 8000000000000 to get 25736787920000.

$$\frac{25736787920000}{1227448863243}+\frac{8+102}{1.07\times 7^{4}}$$

Add 8 and 102 to get 110.

$$\frac{25736787920000}{1227448863243}+\frac{110}{1.07\times 7^{4}}$$

Calculate 7 to the power of 4 and get 2401.

$$\frac{25736787920000}{1227448863243}+\frac{110}{1.07\times 2401}$$

Multiply 1.07 and 2401 to get 2569.07.

$$\frac{25736787920000}{1227448863243}+\frac{110}{2569.07}$$

Expand \frac{110}{2569.07} by multiplying both numerator and the denominator by 100.

$$\frac{25736787920000}{1227448863243}+\frac{11000}{256907}$$

Least common multiple of 1227448863243 and 256907 is 315340205109169401. Convert \frac{25736787920000}{1227448863243} and \frac{11000}{256907} to fractions with denominator 315340205109169401.

$$\frac{6611960974163440000}{315340205109169401}+\frac{13501937495673000}{315340205109169401}$$

Since \frac{6611960974163440000}{315340205109169401} and \frac{13501937495673000}{315340205109169401} have the same denominator, add them by adding their numerators.

$$\frac{6611960974163440000+13501937495673000}{315340205109169401}$$

Add 6611960974163440000 and 13501937495673000 to get 6625462911659113000.

$$\frac{6625462911659113000}{315340205109169401}$$

Factor

$$\frac{71882081 \cdot 92171273 \cdot 2 ^ {3} \cdot 5 ^ {3}}{107 \cdot 3 ^ {3} \cdot 7 ^ {4} \cdot 43 ^ {3} \cdot 83 ^ {3}} = 21\frac{3318604366555904}{3.1534020510916942 \times 10^{17}} = 21.01052388598976$$