Solve frac{2(a^{-2}b^{8})^-3*(ab)^-6}{(2b^-6)^2}

$$\frac { 2 ( a ^ { - 2 } b ^ { 8 } ) ^ { - 3 } \times ( a b ) ^ { - 6 } } { ( 2 b ^ { - 6 } ) ^ { 2 } }$$

Evaluate

$$\frac{1}{2b^{18}}$$

Solution Steps

Expand \left(a^{-2}b^{8}\right)^{-3}.

$$\frac{2\left(a^{-2}\right)^{-3}\left(b^{8}\right)^{-3}\left(ab\right)^{-6}}{\left(2b^{-6}\right)^{2}}$$

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

$$\frac{2a^{6}\left(b^{8}\right)^{-3}\left(ab\right)^{-6}}{\left(2b^{-6}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -3 to get -24.

$$\frac{2a^{6}b^{-24}\left(ab\right)^{-6}}{\left(2b^{-6}\right)^{2}}$$

Expand \left(ab\right)^{-6}.

$$\frac{2a^{6}b^{-24}a^{-6}b^{-6}}{\left(2b^{-6}\right)^{2}}$$

Multiply a^{6} and a^{-6} to get 1.

$$\frac{2b^{-24}b^{-6}}{\left(2b^{-6}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add -24 and -6 to get -30.

$$\frac{2b^{-30}}{\left(2b^{-6}\right)^{2}}$$

Expand \left(2b^{-6}\right)^{2}.

$$\frac{2b^{-30}}{2^{2}\left(b^{-6}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -6 and 2 to get -12.

$$\frac{2b^{-30}}{2^{2}b^{-12}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2b^{-30}}{4b^{-12}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{b^{-30}}{2b^{-12}}$$

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

$$\frac{1}{2b^{18}}$$

Expand

$$\frac{1}{2b^{18}}$$

Solution Steps

Expand \left(a^{-2}b^{8}\right)^{-3}.

$$\frac{2\left(a^{-2}\right)^{-3}\left(b^{8}\right)^{-3}\left(ab\right)^{-6}}{\left(2b^{-6}\right)^{2}}$$

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

$$\frac{2a^{6}\left(b^{8}\right)^{-3}\left(ab\right)^{-6}}{\left(2b^{-6}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -3 to get -24.

$$\frac{2a^{6}b^{-24}\left(ab\right)^{-6}}{\left(2b^{-6}\right)^{2}}$$

Expand \left(ab\right)^{-6}.

$$\frac{2a^{6}b^{-24}a^{-6}b^{-6}}{\left(2b^{-6}\right)^{2}}$$

Multiply a^{6} and a^{-6} to get 1.

$$\frac{2b^{-24}b^{-6}}{\left(2b^{-6}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add -24 and -6 to get -30.

$$\frac{2b^{-30}}{\left(2b^{-6}\right)^{2}}$$

Expand \left(2b^{-6}\right)^{2}.

$$\frac{2b^{-30}}{2^{2}\left(b^{-6}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -6 and 2 to get -12.

$$\frac{2b^{-30}}{2^{2}b^{-12}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2b^{-30}}{4b^{-12}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{b^{-30}}{2b^{-12}}$$

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

$$\frac{1}{2b^{18}}$$