Solve frac{(8ab^{3})^{-2}}{a^{3}b^-1c}div(2b)^3c/(ab^2)^2

$$\frac { ( 8 a b ^ { 3 } ) ^ { - 2 } } { a ^ { 3 } b ^ { - 1 } c } \div \frac { ( 2 b ) ^ { 3 } c } { ( a b ^ { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{1}{512c^{2}a^{3}b^{4}}$$

Solution Steps

Divide \frac{\left(8ab^{3}\right)^{-2}}{a^{3}b^{-1}c} by \frac{\left(2b\right)^{3}c}{\left(ab^{2}\right)^{2}} by multiplying \frac{\left(8ab^{3}\right)^{-2}}{a^{3}b^{-1}c} by the reciprocal of \frac{\left(2b\right)^{3}c}{\left(ab^{2}\right)^{2}}.

$$\frac{\left(8ab^{3}\right)^{-2}\left(ab^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{8^{-2}a^{-2}\left(b^{3}\right)^{-2}\left(ab^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{8^{-2}a^{-2}b^{-6}\left(ab^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{\frac{1}{64}a^{-2}b^{-6}\left(ab^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{\frac{1}{64}a^{-2}b^{-6}a^{2}\left(b^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{\frac{1}{64}a^{-2}b^{-6}a^{2}b^{4}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{\frac{1}{64}b^{-6}b^{4}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

To multiply powers of the same base, add their exponents. Add -6 and 4 to get -2.

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

Multiply c and c to get c^{2}.

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{-1}c^{2}\times \left(2b\right)^{3}}$$

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

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{-1}c^{2}\times 2^{3}b^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{-1}c^{2}\times 8b^{3}}$$

To multiply powers of the same base, add their exponents. Add -1 and 3 to get 2.

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{2}c^{2}\times 8}$$

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

$$\frac{\frac{1}{64}}{8c^{2}a^{3}b^{4}}$$

Express \frac{\frac{1}{64}}{8c^{2}a^{3}b^{4}} as a single fraction.

$$\frac{1}{64\times 8c^{2}a^{3}b^{4}}$$

Multiply 64 and 8 to get 512.

$$\frac{1}{512c^{2}a^{3}b^{4}}$$

Expand

$$\frac{1}{512c^{2}a^{3}b^{4}}$$

Solution Steps

Divide \frac{\left(8ab^{3}\right)^{-2}}{a^{3}b^{-1}c} by \frac{\left(2b\right)^{3}c}{\left(ab^{2}\right)^{2}} by multiplying \frac{\left(8ab^{3}\right)^{-2}}{a^{3}b^{-1}c} by the reciprocal of \frac{\left(2b\right)^{3}c}{\left(ab^{2}\right)^{2}}.

$$\frac{\left(8ab^{3}\right)^{-2}\left(ab^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{8^{-2}a^{-2}\left(b^{3}\right)^{-2}\left(ab^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{8^{-2}a^{-2}b^{-6}\left(ab^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{\frac{1}{64}a^{-2}b^{-6}\left(ab^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{\frac{1}{64}a^{-2}b^{-6}a^{2}\left(b^{2}\right)^{2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{\frac{1}{64}a^{-2}b^{-6}a^{2}b^{4}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

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

$$\frac{\frac{1}{64}b^{-6}b^{4}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

To multiply powers of the same base, add their exponents. Add -6 and 4 to get -2.

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{-1}c\times \left(2b\right)^{3}c}$$

Multiply c and c to get c^{2}.

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{-1}c^{2}\times \left(2b\right)^{3}}$$

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

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{-1}c^{2}\times 2^{3}b^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{-1}c^{2}\times 8b^{3}}$$

To multiply powers of the same base, add their exponents. Add -1 and 3 to get 2.

$$\frac{\frac{1}{64}b^{-2}}{a^{3}b^{2}c^{2}\times 8}$$

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

$$\frac{\frac{1}{64}}{8c^{2}a^{3}b^{4}}$$

Express \frac{\frac{1}{64}}{8c^{2}a^{3}b^{4}} as a single fraction.

$$\frac{1}{64\times 8c^{2}a^{3}b^{4}}$$

Multiply 64 and 8 to get 512.

$$\frac{1}{512c^{2}a^{3}b^{4}}$$