Solve (2ab^3)^-1/(-2a^2b)^-2

$$\frac { ( 2 a b ^ { 3 } ) ^ { - 1 } } { ( - 2 a ^ { 2 } b ) ^ { - 2 } }$$

Evaluate

$$\frac{2a^{3}}{b}$$

Solution Steps

Expand \left(2ab^{3}\right)^{-1}.

$$\frac{2^{-1}a^{-1}\left(b^{3}\right)^{-1}}{\left(-2a^{2}b\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$\frac{2^{-1}a^{-1}b^{-3}}{\left(-2a^{2}b\right)^{-2}}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$\frac{\frac{1}{2}a^{-1}b^{-3}}{\left(-2a^{2}b\right)^{-2}}$$

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

$$\frac{\frac{1}{2}a^{-1}b^{-3}}{\left(-2\right)^{-2}\left(a^{2}\right)^{-2}b^{-2}}$$

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

$$\frac{\frac{1}{2}a^{-1}b^{-3}}{\left(-2\right)^{-2}a^{-4}b^{-2}}$$

Calculate -2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{1}{2}a^{-1}b^{-3}}{\frac{1}{4}a^{-4}b^{-2}}$$

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

$$\frac{\frac{1}{2}b^{-3}a^{3}}{\frac{1}{4}b^{-2}}$$

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

$$\frac{\frac{1}{2}a^{3}}{\frac{1}{4}b^{1}}$$

Calculate b to the power of 1 and get b.

$$\frac{\frac{1}{2}a^{3}}{\frac{1}{4}b}$$

Expand

$$\frac{2a^{3}}{b}$$

Solution Steps

Expand \left(2ab^{3}\right)^{-1}.

$$\frac{2^{-1}a^{-1}\left(b^{3}\right)^{-1}}{\left(-2a^{2}b\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$\frac{2^{-1}a^{-1}b^{-3}}{\left(-2a^{2}b\right)^{-2}}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$\frac{\frac{1}{2}a^{-1}b^{-3}}{\left(-2a^{2}b\right)^{-2}}$$

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

$$\frac{\frac{1}{2}a^{-1}b^{-3}}{\left(-2\right)^{-2}\left(a^{2}\right)^{-2}b^{-2}}$$

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

$$\frac{\frac{1}{2}a^{-1}b^{-3}}{\left(-2\right)^{-2}a^{-4}b^{-2}}$$

Calculate -2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{1}{2}a^{-1}b^{-3}}{\frac{1}{4}a^{-4}b^{-2}}$$

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

$$\frac{\frac{1}{2}b^{-3}a^{3}}{\frac{1}{4}b^{-2}}$$

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

$$\frac{\frac{1}{2}a^{3}}{\frac{1}{4}b^{1}}$$

Calculate b to the power of 1 and get b.

$$\frac{\frac{1}{2}a^{3}}{\frac{1}{4}b}$$