Solve {left(log_{10}({10^2100^3/1000^2*10^4})right)}^4

$${ \left( \log_{ 10 }({ \frac{ { 10 }^{ 2 } { 100 }^{ 3 } }{ { 1000 }^{ 2 } \times { 10 }^{ 4 } } }) \right) }^{ 4 }$$

Evaluate

$$16$$

Solution Steps

Cancel out 10^{2} in both numerator and denominator.

$$\left(\log_{10}\left(\frac{100^{3}}{10^{2}\times 1000^{2}}\right)\right)^{4}$$

Calculate 100 to the power of 3 and get 1000000.

$$\left(\log_{10}\left(\frac{1000000}{10^{2}\times 1000^{2}}\right)\right)^{4}$$

Calculate 10 to the power of 2 and get 100.

$$\left(\log_{10}\left(\frac{1000000}{100\times 1000^{2}}\right)\right)^{4}$$

Calculate 1000 to the power of 2 and get 1000000.

$$\left(\log_{10}\left(\frac{1000000}{100\times 1000000}\right)\right)^{4}$$

Multiply 100 and 1000000 to get 100000000.

$$\left(\log_{10}\left(\frac{1000000}{100000000}\right)\right)^{4}$$

Reduce the fraction \frac{1000000}{100000000} to lowest terms by extracting and canceling out 1000000.

$$\left(\log_{10}\left(\frac{1}{100}\right)\right)^{4}$$

The base 10 logarithm of \frac{1}{100} is -2.

$$\left(-2\right)^{4}$$

Calculate -2 to the power of 4 and get 16.

$$16$$

Factor

$$2^{4}$$