Solve (2/3a^{2}b)^{3}div(a^3b)^2*(-1/2b^2)^2

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

Evaluate

$$\frac{2b^{5}}{27}$$

Solution Steps

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

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

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

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

Calculate \frac{2}{3} to the power of 3 and get \frac{8}{27}.

$$\frac{\frac{8}{27}a^{6}b^{3}}{\left(a^{3}b\right)^{2}}\left(-\frac{1}{2}b^{2}\right)^{2}$$

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

$$\frac{\frac{8}{27}a^{6}b^{3}}{\left(a^{3}\right)^{2}b^{2}}\left(-\frac{1}{2}b^{2}\right)^{2}$$

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

$$\frac{\frac{8}{27}a^{6}b^{3}}{a^{6}b^{2}}\left(-\frac{1}{2}b^{2}\right)^{2}$$

Cancel out b^{2}a^{6} in both numerator and denominator.

$$\frac{8}{27}b\left(-\frac{1}{2}b^{2}\right)^{2}$$

Expand \left(-\frac{1}{2}b^{2}\right)^{2}.

$$\frac{8}{27}b\left(-\frac{1}{2}\right)^{2}\left(b^{2}\right)^{2}$$

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

$$\frac{8}{27}b\left(-\frac{1}{2}\right)^{2}b^{4}$$

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

$$\frac{8}{27}b\times \frac{1}{4}b^{4}$$

Multiply \frac{8}{27} and \frac{1}{4} to get \frac{2}{27}.

$$\frac{2}{27}bb^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{2}{27}b^{5}$$

Expand

$$\frac{2b^{5}}{27}$$

Solution Steps

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

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

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

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

Calculate \frac{2}{3} to the power of 3 and get \frac{8}{27}.

$$\frac{\frac{8}{27}a^{6}b^{3}}{\left(a^{3}b\right)^{2}}\left(-\frac{1}{2}b^{2}\right)^{2}$$

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

$$\frac{\frac{8}{27}a^{6}b^{3}}{\left(a^{3}\right)^{2}b^{2}}\left(-\frac{1}{2}b^{2}\right)^{2}$$

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

$$\frac{\frac{8}{27}a^{6}b^{3}}{a^{6}b^{2}}\left(-\frac{1}{2}b^{2}\right)^{2}$$

Cancel out b^{2}a^{6} in both numerator and denominator.

$$\frac{8}{27}b\left(-\frac{1}{2}b^{2}\right)^{2}$$

Expand \left(-\frac{1}{2}b^{2}\right)^{2}.

$$\frac{8}{27}b\left(-\frac{1}{2}\right)^{2}\left(b^{2}\right)^{2}$$

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

$$\frac{8}{27}b\left(-\frac{1}{2}\right)^{2}b^{4}$$

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

$$\frac{8}{27}b\times \frac{1}{4}b^{4}$$

Multiply \frac{8}{27} and \frac{1}{4} to get \frac{2}{27}.

$$\frac{2}{27}bb^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{2}{27}b^{5}$$