Solve frac{10^{0}e^{-10}}{0!}+frac{10^{1}e^{-10}}{1!}+frac{10^{2}e^{-10}}{2!}+frac{10^{3}e^{-10}}{3!}+frac{10^{4}e^{-10}}{4!}+frac{10^{5}e^{-10}}{5!}+frac{10^{6}e^{-10}}{6!}+10^7e^-10/7!+10^8e^-10/8!

$$\frac { 10 ^ { 0 } e ^ { - 10 } } { 0 ! } + \frac { 10 ^ { 1 } e ^ { - 10 } } { 1 ! } + \frac { 10 ^ { 2 } e ^ { - 10 } } { 2 ! } + \frac { 10 ^ { 3 } e ^ { - 10 } } { 3 ! } + \frac { 10 ^ { 4 } e ^ { - 10 } } { 4 ! } + \frac { 10 ^ { 5 } e ^ { - 10 } } { 5 ! } + \frac { 10 ^ { 6 } e ^ { - 10 } } { 6 ! } + \frac { 10 ^ { 7 } e ^ { - 10 } } { 7 ! } + \frac { 10 ^ { 8 } e ^ { - 10 } } { 8 ! }$$

Evaluate

$$\frac{461843}{63e^{10}}\approx 0.332819679$$

Solution Steps

Calculate 10 to the power of 0 and get 1.

$$\frac{1e^{-10}}{0!}+\frac{10^{1}e^{-10}}{1!}+\frac{10^{2}e^{-10}}{2!}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

The factorial of 0 is 1.

$$\frac{1e^{-10}}{1}+\frac{10^{1}e^{-10}}{1!}+\frac{10^{2}e^{-10}}{2!}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Cancel out 1 and 1.

$$e^{-10}+\frac{10^{1}e^{-10}}{1!}+\frac{10^{2}e^{-10}}{2!}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Calculate 10 to the power of 1 and get 10.

$$e^{-10}+\frac{10e^{-10}}{1!}+\frac{10^{2}e^{-10}}{2!}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

The factorial of 1 is 1.

$$e^{-10}+\frac{10e^{-10}}{1}+\frac{10^{2}e^{-10}}{2!}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Anything divided by one gives itself.

$$e^{-10}+10e^{-10}+\frac{10^{2}e^{-10}}{2!}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Combine e^{-10} and 10e^{-10} to get 11e^{-10}.

$$11e^{-10}+\frac{10^{2}e^{-10}}{2!}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Calculate 10 to the power of 2 and get 100.

$$11e^{-10}+\frac{100e^{-10}}{2!}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

The factorial of 2 is 2.

$$11e^{-10}+\frac{100e^{-10}}{2}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Divide 100e^{-10} by 2 to get 50e^{-10}.

$$11e^{-10}+50e^{-10}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Combine 11e^{-10} and 50e^{-10} to get 61e^{-10}.

$$61e^{-10}+\frac{10^{3}e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Calculate 10 to the power of 3 and get 1000.

$$61e^{-10}+\frac{1000e^{-10}}{3!}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

The factorial of 3 is 6.

$$61e^{-10}+\frac{1000e^{-10}}{6}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Divide 1000e^{-10} by 6 to get \frac{500}{3}e^{-10}.

$$61e^{-10}+\frac{500}{3}e^{-10}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Combine 61e^{-10} and \frac{500}{3}e^{-10} to get \frac{683}{3}e^{-10}.

$$\frac{683}{3}e^{-10}+\frac{10^{4}e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Calculate 10 to the power of 4 and get 10000.

$$\frac{683}{3}e^{-10}+\frac{10000e^{-10}}{4!}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

The factorial of 4 is 24.

$$\frac{683}{3}e^{-10}+\frac{10000e^{-10}}{24}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Divide 10000e^{-10} by 24 to get \frac{1250}{3}e^{-10}.

$$\frac{683}{3}e^{-10}+\frac{1250}{3}e^{-10}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Combine \frac{683}{3}e^{-10} and \frac{1250}{3}e^{-10} to get \frac{1933}{3}e^{-10}.

$$\frac{1933}{3}e^{-10}+\frac{10^{5}e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Calculate 10 to the power of 5 and get 100000.

$$\frac{1933}{3}e^{-10}+\frac{100000e^{-10}}{5!}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

The factorial of 5 is 120.

$$\frac{1933}{3}e^{-10}+\frac{100000e^{-10}}{120}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Divide 100000e^{-10} by 120 to get \frac{2500}{3}e^{-10}.

$$\frac{1933}{3}e^{-10}+\frac{2500}{3}e^{-10}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Combine \frac{1933}{3}e^{-10} and \frac{2500}{3}e^{-10} to get \frac{4433}{3}e^{-10}.

$$\frac{4433}{3}e^{-10}+\frac{10^{6}e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Calculate 10 to the power of 6 and get 1000000.

$$\frac{4433}{3}e^{-10}+\frac{1000000e^{-10}}{6!}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

The factorial of 6 is 720.

$$\frac{4433}{3}e^{-10}+\frac{1000000e^{-10}}{720}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Divide 1000000e^{-10} by 720 to get \frac{12500}{9}e^{-10}.

$$\frac{4433}{3}e^{-10}+\frac{12500}{9}e^{-10}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Combine \frac{4433}{3}e^{-10} and \frac{12500}{9}e^{-10} to get \frac{25799}{9}e^{-10}.

$$\frac{25799}{9}e^{-10}+\frac{10^{7}e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

Calculate 10 to the power of 7 and get 10000000.

$$\frac{25799}{9}e^{-10}+\frac{10000000e^{-10}}{7!}+\frac{10^{8}e^{-10}}{8!}$$

The factorial of 7 is 5040.

$$\frac{25799}{9}e^{-10}+\frac{10000000e^{-10}}{5040}+\frac{10^{8}e^{-10}}{8!}$$

Divide 10000000e^{-10} by 5040 to get \frac{125000}{63}e^{-10}.

$$\frac{25799}{9}e^{-10}+\frac{125000}{63}e^{-10}+\frac{10^{8}e^{-10}}{8!}$$

Combine \frac{25799}{9}e^{-10} and \frac{125000}{63}e^{-10} to get \frac{305593}{63}e^{-10}.

$$\frac{305593}{63}e^{-10}+\frac{10^{8}e^{-10}}{8!}$$

Calculate 10 to the power of 8 and get 100000000.

$$\frac{305593}{63}e^{-10}+\frac{100000000e^{-10}}{8!}$$

The factorial of 8 is 40320.

$$\frac{305593}{63}e^{-10}+\frac{100000000e^{-10}}{40320}$$

Divide 100000000e^{-10} by 40320 to get \frac{156250}{63}e^{-10}.

$$\frac{305593}{63}e^{-10}+\frac{156250}{63}e^{-10}$$

Combine \frac{305593}{63}e^{-10} and \frac{156250}{63}e^{-10} to get \frac{461843}{63}e^{-10}.

$$\frac{461843}{63}e^{-10}$$