Solve frac{(-5)^{6}*(-5)^{8}*5^{7}}{(-5)^{4}*(-5)^2}+(-3/2)^10*(2/3)^10(-1)^2000

$$\frac { ( - 5 ) ^ { 6 } \cdot ( - 5 ) ^ { 8 } \cdot 5 ^ { 7 } } { ( - 5 ) ^ { 4 } \cdot ( - 5 ) ^ { 2 } } + ( - \frac { 3 } { 2 } ) ^ { 10 } \cdot ( \frac { 2 } { 3 } ) ^ { 10 } ( - 1 ) ^ { 2000 }$$

Evaluate

$$30517578126$$

Solution Steps

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

$$\frac{\left(-5\right)^{14}\times 5^{7}}{\left(-5\right)^{4}\left(-5\right)^{2}}+\left(-\frac{3}{2}\right)^{10}\times \left(\frac{2}{3}\right)^{10}\left(-1\right)^{2000}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\left(-5\right)^{14}\times 5^{7}}{\left(-5\right)^{6}}+\left(-\frac{3}{2}\right)^{10}\times \left(\frac{2}{3}\right)^{10}\left(-1\right)^{2000}$$

Cancel out \left(-5\right)^{6} in both numerator and denominator.

$$\left(-5\right)^{8}\times 5^{7}+\left(-\frac{3}{2}\right)^{10}\times \left(\frac{2}{3}\right)^{10}\left(-1\right)^{2000}$$

Calculate -\frac{3}{2} to the power of 10 and get \frac{59049}{1024}.

$$\left(-5\right)^{8}\times 5^{7}+\frac{59049}{1024}\times \left(\frac{2}{3}\right)^{10}\left(-1\right)^{2000}$$

Calculate \frac{2}{3} to the power of 10 and get \frac{1024}{59049}.

$$\left(-5\right)^{8}\times 5^{7}+\frac{59049}{1024}\times \frac{1024}{59049}\left(-1\right)^{2000}$$

Cancel out \frac{59049}{1024} and its reciprocal \frac{1024}{59049}.

$$\left(-5\right)^{8}\times 5^{7}+\left(-1\right)^{2000}$$

Calculate -1 to the power of 2000 and get 1.

$$\left(-5\right)^{8}\times 5^{7}+1$$

Calculate -5 to the power of 8 and get 390625.

$$390625\times 5^{7}+1$$

Calculate 5 to the power of 7 and get 78125.

$$390625\times 78125+1$$

Multiply 390625 and 78125 to get 30517578125.

$$30517578125+1$$

Add 30517578125 and 1 to get 30517578126.

$$30517578126$$

Factor

$$2\times 3^{2}\times 7\times 61\times 521\times 7621$$