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

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

Evaluate

$$-\frac{b^{3}}{16}$$

Solution Steps

Calculate -\frac{a^{3}}{2b} to the power of 2 and get \left(\frac{a^{3}}{2b}\right)^{2}.

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

To raise \frac{a^{3}}{2b} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

Express \frac{\frac{a^{6}}{\left(2b\right)^{2}}}{\left(-\frac{a^{2}}{b}\right)^{3}} as a single fraction.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

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

Multiply 4 and -1 to get -4.

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

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

$$\frac{a^{6}}{-4b^{2}\times \frac{a^{6}}{b^{3}}}\times \frac{b^{2}}{4}$$

Express -4\times \frac{a^{6}}{b^{3}} as a single fraction.

$$\frac{a^{6}}{\frac{-4a^{6}}{b^{3}}b^{2}}\times \frac{b^{2}}{4}$$

Express \frac{-4a^{6}}{b^{3}}b^{2} as a single fraction.

$$\frac{a^{6}}{\frac{-4a^{6}b^{2}}{b^{3}}}\times \frac{b^{2}}{4}$$

Cancel out b^{2} in both numerator and denominator.

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

Divide a^{6} by \frac{-4a^{6}}{b} by multiplying a^{6} by the reciprocal of \frac{-4a^{6}}{b}.

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

Cancel out a^{6} in both numerator and denominator.

$$\frac{b}{-4}\times \frac{b^{2}}{4}$$

Multiply \frac{b}{-4} times \frac{b^{2}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{bb^{2}}{-4\times 4}$$

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

$$\frac{b^{3}}{-4\times 4}$$

Multiply -4 and 4 to get -16.

$$\frac{b^{3}}{-16}$$

Expand

$$-\frac{b^{3}}{16}$$

Solution Steps

Calculate -\frac{a^{3}}{2b} to the power of 2 and get \left(\frac{a^{3}}{2b}\right)^{2}.

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

To raise \frac{a^{3}}{2b} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

Express \frac{\frac{a^{6}}{\left(2b\right)^{2}}}{\left(-\frac{a^{2}}{b}\right)^{3}} as a single fraction.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

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

Multiply 4 and -1 to get -4.

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

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

$$\frac{a^{6}}{-4b^{2}\times \frac{a^{6}}{b^{3}}}\times \frac{b^{2}}{4}$$

Express -4\times \frac{a^{6}}{b^{3}} as a single fraction.

$$\frac{a^{6}}{\frac{-4a^{6}}{b^{3}}b^{2}}\times \frac{b^{2}}{4}$$

Express \frac{-4a^{6}}{b^{3}}b^{2} as a single fraction.

$$\frac{a^{6}}{\frac{-4a^{6}b^{2}}{b^{3}}}\times \frac{b^{2}}{4}$$

Cancel out b^{2} in both numerator and denominator.

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

Divide a^{6} by \frac{-4a^{6}}{b} by multiplying a^{6} by the reciprocal of \frac{-4a^{6}}{b}.

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

Cancel out a^{6} in both numerator and denominator.

$$\frac{b}{-4}\times \frac{b^{2}}{4}$$

Multiply \frac{b}{-4} times \frac{b^{2}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{bb^{2}}{-4\times 4}$$

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

$$\frac{b^{3}}{-4\times 4}$$

Multiply -4 and 4 to get -16.

$$\frac{b^{3}}{-16}$$