Solve [-3/2a^{3}b^{2}]^{4}*[a^2/3a^2b^5]^3

$$[ - \frac { 3 } { 2 } a ^ { 3 } b ^ { 2 } ] ^ { 4 } \cdot [ \frac { a ^ { 2 } } { 3 } a ^ { 2 } b ^ { 5 } ] ^ { 3 }$$

Evaluate

$$\frac{3b^{23}a^{24}}{16}$$

Solution Steps

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

$$\left(-\frac{3}{2}\right)^{4}\left(a^{3}\right)^{4}\left(b^{2}\right)^{4}\times \left(\frac{a^{2}}{3}a^{2}b^{5}\right)^{3}$$

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

$$\left(-\frac{3}{2}\right)^{4}a^{12}\left(b^{2}\right)^{4}\times \left(\frac{a^{2}}{3}a^{2}b^{5}\right)^{3}$$

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

$$\left(-\frac{3}{2}\right)^{4}a^{12}b^{8}\times \left(\frac{a^{2}}{3}a^{2}b^{5}\right)^{3}$$

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

$$\frac{81}{16}a^{12}b^{8}\times \left(\frac{a^{2}}{3}a^{2}b^{5}\right)^{3}$$

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

$$\frac{81}{16}a^{12}b^{8}\times \left(\frac{a^{2}a^{2}}{3}b^{5}\right)^{3}$$

Express \frac{a^{2}a^{2}}{3}b^{5} as a single fraction.

$$\frac{81}{16}a^{12}b^{8}\times \left(\frac{a^{2}a^{2}b^{5}}{3}\right)^{3}$$

To raise \frac{a^{2}a^{2}b^{5}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{81}{16}a^{12}b^{8}\times \frac{\left(a^{2}a^{2}b^{5}\right)^{3}}{3^{3}}$$

Multiply \frac{81}{16} times \frac{\left(a^{2}a^{2}b^{5}\right)^{3}}{3^{3}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{81\left(a^{2}a^{2}b^{5}\right)^{3}}{16\times 3^{3}}a^{12}b^{8}$$

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

$$\frac{81\left(a^{4}b^{5}\right)^{3}}{16\times 3^{3}}a^{12}b^{8}$$

Expand \left(a^{4}b^{5}\right)^{3}.

$$\frac{81\left(a^{4}\right)^{3}\left(b^{5}\right)^{3}}{16\times 3^{3}}a^{12}b^{8}$$

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

$$\frac{81a^{12}\left(b^{5}\right)^{3}}{16\times 3^{3}}a^{12}b^{8}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{81a^{12}b^{15}}{16\times 3^{3}}a^{12}b^{8}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{81a^{12}b^{15}}{16\times 27}a^{12}b^{8}$$

Multiply 16 and 27 to get 432.

$$\frac{81a^{12}b^{15}}{432}a^{12}b^{8}$$

Divide 81a^{12}b^{15} by 432 to get \frac{3}{16}a^{12}b^{15}.

$$\frac{3}{16}a^{12}b^{15}a^{12}b^{8}$$

To multiply powers of the same base, add their exponents. Add 12 and 12 to get 24.

$$\frac{3}{16}a^{24}b^{15}b^{8}$$

To multiply powers of the same base, add their exponents. Add 15 and 8 to get 23.

$$\frac{3}{16}a^{24}b^{23}$$

Expand

$$\frac{3b^{23}a^{24}}{16}$$

Solution Steps

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

$$\left(-\frac{3}{2}\right)^{4}\left(a^{3}\right)^{4}\left(b^{2}\right)^{4}\times \left(\frac{a^{2}}{3}a^{2}b^{5}\right)^{3}$$

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

$$\left(-\frac{3}{2}\right)^{4}a^{12}\left(b^{2}\right)^{4}\times \left(\frac{a^{2}}{3}a^{2}b^{5}\right)^{3}$$

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

$$\left(-\frac{3}{2}\right)^{4}a^{12}b^{8}\times \left(\frac{a^{2}}{3}a^{2}b^{5}\right)^{3}$$

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

$$\frac{81}{16}a^{12}b^{8}\times \left(\frac{a^{2}}{3}a^{2}b^{5}\right)^{3}$$

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

$$\frac{81}{16}a^{12}b^{8}\times \left(\frac{a^{2}a^{2}}{3}b^{5}\right)^{3}$$

Express \frac{a^{2}a^{2}}{3}b^{5} as a single fraction.

$$\frac{81}{16}a^{12}b^{8}\times \left(\frac{a^{2}a^{2}b^{5}}{3}\right)^{3}$$

To raise \frac{a^{2}a^{2}b^{5}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{81}{16}a^{12}b^{8}\times \frac{\left(a^{2}a^{2}b^{5}\right)^{3}}{3^{3}}$$

Multiply \frac{81}{16} times \frac{\left(a^{2}a^{2}b^{5}\right)^{3}}{3^{3}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{81\left(a^{2}a^{2}b^{5}\right)^{3}}{16\times 3^{3}}a^{12}b^{8}$$

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

$$\frac{81\left(a^{4}b^{5}\right)^{3}}{16\times 3^{3}}a^{12}b^{8}$$

Expand \left(a^{4}b^{5}\right)^{3}.

$$\frac{81\left(a^{4}\right)^{3}\left(b^{5}\right)^{3}}{16\times 3^{3}}a^{12}b^{8}$$

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

$$\frac{81a^{12}\left(b^{5}\right)^{3}}{16\times 3^{3}}a^{12}b^{8}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{81a^{12}b^{15}}{16\times 3^{3}}a^{12}b^{8}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{81a^{12}b^{15}}{16\times 27}a^{12}b^{8}$$

Multiply 16 and 27 to get 432.

$$\frac{81a^{12}b^{15}}{432}a^{12}b^{8}$$

Divide 81a^{12}b^{15} by 432 to get \frac{3}{16}a^{12}b^{15}.

$$\frac{3}{16}a^{12}b^{15}a^{12}b^{8}$$

To multiply powers of the same base, add their exponents. Add 12 and 12 to get 24.

$$\frac{3}{16}a^{24}b^{15}b^{8}$$

To multiply powers of the same base, add their exponents. Add 15 and 8 to get 23.

$$\frac{3}{16}a^{24}b^{23}$$