Solve frac{(-2k^{-3})^{2}(6k^2)^4}{(9k^4)^2}

$$\frac { ( - 2 k ^ { - 3 } ) ^ { 2 } ( 6 k ^ { 2 } ) ^ { 4 } } { ( 9 k ^ { 4 } ) ^ { 2 } }$$

Evaluate

$$\frac{64}{k^{6}}$$

Solution Steps

Expand \left(-2k^{-3}\right)^{2}.

$$\frac{\left(-2\right)^{2}\left(k^{-3}\right)^{2}\times \left(6k^{2}\right)^{4}}{\left(9k^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\frac{\left(-2\right)^{2}k^{-6}\times \left(6k^{2}\right)^{4}}{\left(9k^{4}\right)^{2}}$$

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

$$\frac{4k^{-6}\times \left(6k^{2}\right)^{4}}{\left(9k^{4}\right)^{2}}$$

Expand \left(6k^{2}\right)^{4}.

$$\frac{4k^{-6}\times 6^{4}\left(k^{2}\right)^{4}}{\left(9k^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{4k^{-6}\times 6^{4}k^{8}}{\left(9k^{4}\right)^{2}}$$

Calculate 6 to the power of 4 and get 1296.

$$\frac{4k^{-6}\times 1296k^{8}}{\left(9k^{4}\right)^{2}}$$

Multiply 4 and 1296 to get 5184.

$$\frac{5184k^{-6}k^{8}}{\left(9k^{4}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add -6 and 8 to get 2.

$$\frac{5184k^{2}}{\left(9k^{4}\right)^{2}}$$

Expand \left(9k^{4}\right)^{2}.

$$\frac{5184k^{2}}{9^{2}\left(k^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{5184k^{2}}{9^{2}k^{8}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{5184k^{2}}{81k^{8}}$$

Cancel out 81k^{2} in both numerator and denominator.

$$\frac{64}{k^{6}}$$

Expand

$$\frac{64}{k^{6}}$$

Solution Steps

Expand \left(-2k^{-3}\right)^{2}.

$$\frac{\left(-2\right)^{2}\left(k^{-3}\right)^{2}\times \left(6k^{2}\right)^{4}}{\left(9k^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\frac{\left(-2\right)^{2}k^{-6}\times \left(6k^{2}\right)^{4}}{\left(9k^{4}\right)^{2}}$$

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

$$\frac{4k^{-6}\times \left(6k^{2}\right)^{4}}{\left(9k^{4}\right)^{2}}$$

Expand \left(6k^{2}\right)^{4}.

$$\frac{4k^{-6}\times 6^{4}\left(k^{2}\right)^{4}}{\left(9k^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{4k^{-6}\times 6^{4}k^{8}}{\left(9k^{4}\right)^{2}}$$

Calculate 6 to the power of 4 and get 1296.

$$\frac{4k^{-6}\times 1296k^{8}}{\left(9k^{4}\right)^{2}}$$

Multiply 4 and 1296 to get 5184.

$$\frac{5184k^{-6}k^{8}}{\left(9k^{4}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add -6 and 8 to get 2.

$$\frac{5184k^{2}}{\left(9k^{4}\right)^{2}}$$

Expand \left(9k^{4}\right)^{2}.

$$\frac{5184k^{2}}{9^{2}\left(k^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{5184k^{2}}{9^{2}k^{8}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{5184k^{2}}{81k^{8}}$$

Cancel out 81k^{2} in both numerator and denominator.

$$\frac{64}{k^{6}}$$