Solve [frac{(-30)^{6}(-30)^{4}(-30)^{8}(-4)^{2}}{(-4)^15(-4)^2(-30)^-6}]^4

$$[ \frac { ( - 30 ) ^ { 6 } ( - 30 ) ^ { 4 } ( - 30 ) ^ { 8 } ( - 4 ) ^ { 2 } } { ( - 4 ) ^ { 15 } ( - 4 ) ^ { 2 } ( - 30 ) ^ { - 6 } } ] ^ { 4 }$$

Evaluate

$$\frac{80308380747696837016746608902131688100817361112568842158825040963704964581959533376220861100591719150543212890625}{16777216}\approx 4.786752507 \cdot 10^{105}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 6 and 4 to get 10.

$$\left(\frac{\left(-30\right)^{10}\left(-30\right)^{8}\left(-4\right)^{2}}{\left(-4\right)^{15}\left(-4\right)^{2}\left(-30\right)^{-6}}\right)^{4}$$

To multiply powers of the same base, add their exponents. Add 10 and 8 to get 18.

$$\left(\frac{\left(-30\right)^{18}\left(-4\right)^{2}}{\left(-4\right)^{15}\left(-4\right)^{2}\left(-30\right)^{-6}}\right)^{4}$$

To multiply powers of the same base, add their exponents. Add 15 and 2 to get 17.

$$\left(\frac{\left(-30\right)^{18}\left(-4\right)^{2}}{\left(-4\right)^{17}\left(-30\right)^{-6}}\right)^{4}$$

Cancel out \left(-4\right)^{2} in both numerator and denominator.

$$\left(\frac{\left(-30\right)^{18}}{\left(-30\right)^{-6}\left(-4\right)^{15}}\right)^{4}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\left(\frac{\left(-30\right)^{24}}{\left(-4\right)^{15}}\right)^{4}$$

Calculate -30 to the power of 24 and get 282429536481000000000000000000000000.

$$\left(\frac{282429536481000000000000000000000000}{\left(-4\right)^{15}}\right)^{4}$$

Calculate -4 to the power of 15 and get -1073741824.

$$\left(\frac{282429536481000000000000000000000000}{-1073741824}\right)^{4}$$

Reduce the fraction \frac{282429536481000000000000000000000000}{-1073741824} to lowest terms by extracting and canceling out 16777216.

$$\left(-\frac{16834112196028232574462890625}{64}\right)^{4}$$

Calculate -\frac{16834112196028232574462890625}{64} to the power of 4 and get \frac{80308380747696837016746608902131688100817361112568842158825040963704964581959533376220861100591719150543212890625}{16777216}.

$$\frac{80308380747696837016746608902131688100817361112568842158825040963704964581959533376220861100591719150543212890625}{16777216}$$

Factor

$$\frac{3 ^ {96} \cdot 5 ^ {96}}{2 ^ {24}} = 4.78675250695329 \times 10^{105}$$