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

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

Evaluate

$$\frac{b^{2}a^{8}}{2}$$

Solution Steps

To raise \frac{1}{ab} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

Express \left(-\frac{b^{2}}{2a}\right)a^{3} as a single fraction.

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

Cancel out a in both numerator and denominator.

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

Express \frac{-a^{2}b^{2}}{2}b^{3} as a single fraction.

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

Calculate 1 to the power of 3 and get 1.

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

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{\frac{-a^{2}b^{5}}{2}}{\left(-\frac{b}{a^{2}}\right)^{3}\times 1}$$

Express \frac{\frac{-a^{2}b^{5}}{2}}{\left(-\frac{b}{a^{2}}\right)^{3}\times 1} as a single fraction.

$$\frac{-a^{2}b^{5}}{2\left(-\frac{b}{a^{2}}\right)^{3}\times 1}$$

Multiply 2 and 1 to get 2.

$$\frac{-a^{2}b^{5}}{2\left(-\frac{b}{a^{2}}\right)^{3}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{a^{2}b^{5}}{-2\left(-\frac{b}{a^{2}}\right)^{3}}$$

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

$$\frac{a^{2}b^{5}}{-2\left(-1\right)^{3}\times \left(\frac{b}{a^{2}}\right)^{3}}$$

Calculate -1 to the power of 3 and get -1.

$$\frac{a^{2}b^{5}}{-2\left(-1\right)\times \left(\frac{b}{a^{2}}\right)^{3}}$$

To raise \frac{b}{a^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{a^{2}b^{5}}{-2\left(-1\right)\times \frac{b^{3}}{\left(a^{2}\right)^{3}}}$$

Multiply -2 and -1 to get 2.

$$\frac{a^{2}b^{5}}{2\times \frac{b^{3}}{\left(a^{2}\right)^{3}}}$$

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

$$\frac{a^{2}b^{5}}{2\times \frac{b^{3}}{a^{6}}}$$

Express 2\times \frac{b^{3}}{a^{6}} as a single fraction.

$$\frac{a^{2}b^{5}}{\frac{2b^{3}}{a^{6}}}$$

Divide a^{2}b^{5} by \frac{2b^{3}}{a^{6}} by multiplying a^{2}b^{5} by the reciprocal of \frac{2b^{3}}{a^{6}}.

$$\frac{a^{2}b^{5}a^{6}}{2b^{3}}$$

Cancel out b^{3} in both numerator and denominator.

$$\frac{a^{2}b^{2}a^{6}}{2}$$

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

$$\frac{a^{8}b^{2}}{2}$$

Expand

$$\frac{b^{2}a^{8}}{2}$$

Solution Steps

To raise \frac{1}{ab} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

Express \left(-\frac{b^{2}}{2a}\right)a^{3} as a single fraction.

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

Cancel out a in both numerator and denominator.

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

Express \frac{-a^{2}b^{2}}{2}b^{3} as a single fraction.

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

Calculate 1 to the power of 3 and get 1.

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

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{\frac{-a^{2}b^{5}}{2}}{\left(-\frac{b}{a^{2}}\right)^{3}\times 1}$$

Express \frac{\frac{-a^{2}b^{5}}{2}}{\left(-\frac{b}{a^{2}}\right)^{3}\times 1} as a single fraction.

$$\frac{-a^{2}b^{5}}{2\left(-\frac{b}{a^{2}}\right)^{3}\times 1}$$

Multiply 2 and 1 to get 2.

$$\frac{-a^{2}b^{5}}{2\left(-\frac{b}{a^{2}}\right)^{3}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{a^{2}b^{5}}{-2\left(-\frac{b}{a^{2}}\right)^{3}}$$

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

$$\frac{a^{2}b^{5}}{-2\left(-1\right)^{3}\times \left(\frac{b}{a^{2}}\right)^{3}}$$

Calculate -1 to the power of 3 and get -1.

$$\frac{a^{2}b^{5}}{-2\left(-1\right)\times \left(\frac{b}{a^{2}}\right)^{3}}$$

To raise \frac{b}{a^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{a^{2}b^{5}}{-2\left(-1\right)\times \frac{b^{3}}{\left(a^{2}\right)^{3}}}$$

Multiply -2 and -1 to get 2.

$$\frac{a^{2}b^{5}}{2\times \frac{b^{3}}{\left(a^{2}\right)^{3}}}$$

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

$$\frac{a^{2}b^{5}}{2\times \frac{b^{3}}{a^{6}}}$$

Express 2\times \frac{b^{3}}{a^{6}} as a single fraction.

$$\frac{a^{2}b^{5}}{\frac{2b^{3}}{a^{6}}}$$

Divide a^{2}b^{5} by \frac{2b^{3}}{a^{6}} by multiplying a^{2}b^{5} by the reciprocal of \frac{2b^{3}}{a^{6}}.

$$\frac{a^{2}b^{5}a^{6}}{2b^{3}}$$

Cancel out b^{3} in both numerator and denominator.

$$\frac{a^{2}b^{2}a^{6}}{2}$$

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

$$\frac{a^{8}b^{2}}{2}$$