Solve frac{3(a^{2}b^{-4})^{2}}{(2ab^{2})^2}div(ab)^-2/(3a^-2b)^2

$$\frac { 3 ( a ^ { 2 } b ^ { - 4 } ) ^ { 2 } } { ( 2 a b ^ { 2 } ) ^ { 2 } } \div \frac { ( a b ) ^ { - 2 } } { ( 3 a ^ { - 2 } b ) ^ { 2 } }$$

Evaluate

$$\frac{27}{4b^{8}}$$

Solution Steps

Divide \frac{3\left(a^{2}b^{-4}\right)^{2}}{\left(2ab^{2}\right)^{2}} by \frac{\left(ab\right)^{-2}}{\left(3a^{-2}b\right)^{2}} by multiplying \frac{3\left(a^{2}b^{-4}\right)^{2}}{\left(2ab^{2}\right)^{2}} by the reciprocal of \frac{\left(ab\right)^{-2}}{\left(3a^{-2}b\right)^{2}}.

$$\frac{3\left(a^{2}b^{-4}\right)^{2}\times \left(3a^{-2}b\right)^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Expand \left(a^{2}b^{-4}\right)^{2}.

$$\frac{3\left(a^{2}\right)^{2}\left(b^{-4}\right)^{2}\times \left(3a^{-2}b\right)^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{3a^{4}\left(b^{-4}\right)^{2}\times \left(3a^{-2}b\right)^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{3a^{4}b^{-8}\times \left(3a^{-2}b\right)^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{3a^{4}b^{-8}\times 3^{2}\left(a^{-2}\right)^{2}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{3a^{4}b^{-8}\times 3^{2}a^{-4}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3a^{4}b^{-8}\times 9a^{-4}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Multiply 3 and 9 to get 27.

$$\frac{27a^{4}b^{-8}a^{-4}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Multiply a^{4} and a^{-4} to get 1.

$$\frac{27b^{-8}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{27b^{-6}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Expand \left(2ab^{2}\right)^{2}.

$$\frac{27b^{-6}}{2^{2}a^{2}\left(b^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{27b^{-6}}{2^{2}a^{2}b^{4}\left(ab\right)^{-2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{27b^{-6}}{4a^{2}b^{4}\left(ab\right)^{-2}}$$

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

$$\frac{27b^{-6}}{4a^{2}b^{4}a^{-2}b^{-2}}$$

Multiply a^{2} and a^{-2} to get 1.

$$\frac{27b^{-6}}{4b^{4}b^{-2}}$$

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

$$\frac{27b^{-6}}{4b^{2}}$$

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

$$\frac{27}{4b^{8}}$$

Expand

$$\frac{27}{4b^{8}}$$

Solution Steps

Divide \frac{3\left(a^{2}b^{-4}\right)^{2}}{\left(2ab^{2}\right)^{2}} by \frac{\left(ab\right)^{-2}}{\left(3a^{-2}b\right)^{2}} by multiplying \frac{3\left(a^{2}b^{-4}\right)^{2}}{\left(2ab^{2}\right)^{2}} by the reciprocal of \frac{\left(ab\right)^{-2}}{\left(3a^{-2}b\right)^{2}}.

$$\frac{3\left(a^{2}b^{-4}\right)^{2}\times \left(3a^{-2}b\right)^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Expand \left(a^{2}b^{-4}\right)^{2}.

$$\frac{3\left(a^{2}\right)^{2}\left(b^{-4}\right)^{2}\times \left(3a^{-2}b\right)^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{3a^{4}\left(b^{-4}\right)^{2}\times \left(3a^{-2}b\right)^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{3a^{4}b^{-8}\times \left(3a^{-2}b\right)^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{3a^{4}b^{-8}\times 3^{2}\left(a^{-2}\right)^{2}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{3a^{4}b^{-8}\times 3^{2}a^{-4}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3a^{4}b^{-8}\times 9a^{-4}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Multiply 3 and 9 to get 27.

$$\frac{27a^{4}b^{-8}a^{-4}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Multiply a^{4} and a^{-4} to get 1.

$$\frac{27b^{-8}b^{2}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{27b^{-6}}{\left(2ab^{2}\right)^{2}\left(ab\right)^{-2}}$$

Expand \left(2ab^{2}\right)^{2}.

$$\frac{27b^{-6}}{2^{2}a^{2}\left(b^{2}\right)^{2}\left(ab\right)^{-2}}$$

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

$$\frac{27b^{-6}}{2^{2}a^{2}b^{4}\left(ab\right)^{-2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{27b^{-6}}{4a^{2}b^{4}\left(ab\right)^{-2}}$$

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

$$\frac{27b^{-6}}{4a^{2}b^{4}a^{-2}b^{-2}}$$

Multiply a^{2} and a^{-2} to get 1.

$$\frac{27b^{-6}}{4b^{4}b^{-2}}$$

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

$$\frac{27b^{-6}}{4b^{2}}$$

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

$$\frac{27}{4b^{8}}$$