Solve [-3/2a^{3}b^{2}]^4*(-2/3a^2b^3)^3

$$[ - \frac { 3 } { 2 } a ^ { 3 } b ^ { 2 } ] ^ { 4 } \cdot ( - \frac { 2 } { 3 } a ^ { 2 } b ^ { 3 } ) ^ { 3 }$$

Evaluate

$$-\frac{3b^{17}a^{18}}{2}$$

Solution Steps

Expand \left(-\frac{3}{2}a^{3}b^{2}\right)^{4}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$\left(-\frac{3}{2}\right)^{4}a^{12}\left(b^{2}\right)^{4}\left(-\frac{2}{3}a^{2}b^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\left(-\frac{3}{2}\right)^{4}a^{12}b^{8}\left(-\frac{2}{3}a^{2}b^{3}\right)^{3}$$

Calculate -\frac{3}{2} to the power of 4 and get \frac{81}{16}.

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{2}{3}a^{2}b^{3}\right)^{3}$$

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

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{2}{3}\right)^{3}\left(a^{2}\right)^{3}\left(b^{3}\right)^{3}$$

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

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{2}{3}\right)^{3}a^{6}\left(b^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{2}{3}\right)^{3}a^{6}b^{9}$$

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

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{8}{27}\right)a^{6}b^{9}$$

Multiply \frac{81}{16} and -\frac{8}{27} to get -\frac{3}{2}.

$$-\frac{3}{2}a^{12}b^{8}a^{6}b^{9}$$

To multiply powers of the same base, add their exponents. Add 12 and 6 to get 18.

$$-\frac{3}{2}a^{18}b^{8}b^{9}$$

To multiply powers of the same base, add their exponents. Add 8 and 9 to get 17.

$$-\frac{3}{2}a^{18}b^{17}$$

Expand

$$-\frac{3b^{17}a^{18}}{2}$$

Solution Steps

Expand \left(-\frac{3}{2}a^{3}b^{2}\right)^{4}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$\left(-\frac{3}{2}\right)^{4}a^{12}\left(b^{2}\right)^{4}\left(-\frac{2}{3}a^{2}b^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\left(-\frac{3}{2}\right)^{4}a^{12}b^{8}\left(-\frac{2}{3}a^{2}b^{3}\right)^{3}$$

Calculate -\frac{3}{2} to the power of 4 and get \frac{81}{16}.

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{2}{3}a^{2}b^{3}\right)^{3}$$

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

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{2}{3}\right)^{3}\left(a^{2}\right)^{3}\left(b^{3}\right)^{3}$$

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

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{2}{3}\right)^{3}a^{6}\left(b^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{2}{3}\right)^{3}a^{6}b^{9}$$

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

$$\frac{81}{16}a^{12}b^{8}\left(-\frac{8}{27}\right)a^{6}b^{9}$$

Multiply \frac{81}{16} and -\frac{8}{27} to get -\frac{3}{2}.

$$-\frac{3}{2}a^{12}b^{8}a^{6}b^{9}$$

To multiply powers of the same base, add their exponents. Add 12 and 6 to get 18.

$$-\frac{3}{2}a^{18}b^{8}b^{9}$$

To multiply powers of the same base, add their exponents. Add 8 and 9 to get 17.

$$-\frac{3}{2}a^{18}b^{17}$$