Solve {[(6*6^{5}*6^{8})^{5}:(60^{9}:10^{9})^{7}]^{6}:(6^{5})^{8}}^{9}:(2^{5}*8^3:4^4)^3*125^6

$$\{ [ ( 6 \cdot 6 ^ { 5 } \cdot 6 ^ { 8 } ) ^ { 5 } : ( 60 ^ { 9 } : 10 ^ { 9 } ) ^ { 7 } ] ^ { 6 } : ( 6 ^ { 5 } ) ^ { 8 } \} ^ { 9 } : ( 2 ^ { 5 } \cdot 8 ^ { 3 } : 4 ^ { 4 } ) ^ { 3 } \cdot 125 ^ { 6 }$$

Evaluate

$$1477891880035400390625$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 5 to get 6.

$$\frac{\left(\frac{\left(\frac{\left(6^{6}\times 6^{8}\right)^{5}}{\left(\frac{60^{9}}{10^{9}}\right)^{7}}\right)^{6}}{\left(6^{5}\right)^{8}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

To multiply powers of the same base, add their exponents. Add 6 and 8 to get 14.

$$\frac{\left(\frac{\left(\frac{\left(6^{14}\right)^{5}}{\left(\frac{60^{9}}{10^{9}}\right)^{7}}\right)^{6}}{\left(6^{5}\right)^{8}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

To raise a power to another power, multiply the exponents. Multiply 14 and 5 to get 70.

$$\frac{\left(\frac{\left(\frac{6^{70}}{\left(\frac{60^{9}}{10^{9}}\right)^{7}}\right)^{6}}{\left(6^{5}\right)^{8}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 8 to get 40.

$$\frac{\left(\frac{\left(\frac{6^{70}}{\left(\frac{60^{9}}{10^{9}}\right)^{7}}\right)^{6}}{6^{40}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 6 to the power of 70 and get 2955204414547681244658707659790455381671329323051646976.

$$\frac{\left(\frac{\left(\frac{2955204414547681244658707659790455381671329323051646976}{\left(\frac{60^{9}}{10^{9}}\right)^{7}}\right)^{6}}{6^{40}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 60 to the power of 9 and get 10077696000000000.

$$\frac{\left(\frac{\left(\frac{2955204414547681244658707659790455381671329323051646976}{\left(\frac{10077696000000000}{10^{9}}\right)^{7}}\right)^{6}}{6^{40}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 10 to the power of 9 and get 1000000000.

$$\frac{\left(\frac{\left(\frac{2955204414547681244658707659790455381671329323051646976}{\left(\frac{10077696000000000}{1000000000}\right)^{7}}\right)^{6}}{6^{40}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Divide 10077696000000000 by 1000000000 to get 10077696.

$$\frac{\left(\frac{\left(\frac{2955204414547681244658707659790455381671329323051646976}{10077696^{7}}\right)^{6}}{6^{40}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 10077696 to the power of 7 and get 10556714443828879617693714491135314434982743638016.

$$\frac{\left(\frac{\left(\frac{2955204414547681244658707659790455381671329323051646976}{10556714443828879617693714491135314434982743638016}\right)^{6}}{6^{40}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Divide 2955204414547681244658707659790455381671329323051646976 by 10556714443828879617693714491135314434982743638016 to get 279936.

$$\frac{\left(\frac{279936^{6}}{6^{40}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 279936 to the power of 6 and get 481229803398374426442198455156736.

$$\frac{\left(\frac{481229803398374426442198455156736}{6^{40}}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 6 to the power of 40 and get 13367494538843734067838845976576.

$$\frac{\left(\frac{481229803398374426442198455156736}{13367494538843734067838845976576}\right)^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Divide 481229803398374426442198455156736 by 13367494538843734067838845976576 to get 36.

$$\frac{36^{9}}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 36 to the power of 9 and get 101559956668416.

$$\frac{101559956668416}{\left(\frac{2^{5}\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{101559956668416}{\left(\frac{32\times 8^{3}}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 8 to the power of 3 and get 512.

$$\frac{101559956668416}{\left(\frac{32\times 512}{4^{4}}\right)^{3}}\times 125^{6}$$

Multiply 32 and 512 to get 16384.

$$\frac{101559956668416}{\left(\frac{16384}{4^{4}}\right)^{3}}\times 125^{6}$$

Calculate 4 to the power of 4 and get 256.

$$\frac{101559956668416}{\left(\frac{16384}{256}\right)^{3}}\times 125^{6}$$

Divide 16384 by 256 to get 64.

$$\frac{101559956668416}{64^{3}}\times 125^{6}$$

Calculate 64 to the power of 3 and get 262144.

$$\frac{101559956668416}{262144}\times 125^{6}$$

Divide 101559956668416 by 262144 to get 387420489.

$$387420489\times 125^{6}$$

Calculate 125 to the power of 6 and get 3814697265625.

$$387420489\times 3814697265625$$

Multiply 387420489 and 3814697265625 to get 1477891880035400390625.

$$1477891880035400390625$$

Factor

$$3^{18}\times 5^{18}$$