Solve =324(-1/4a^{2}b^{4})^{3}+(+1/2ab^{2})^{6}+(-18/5a^2b^7)(+10/9a^4b^5)

$$= 324 ( - \frac { 1 } { 4 } a ^ { 2 } b ^ { 4 } ) ^ { 3 } + ( + \frac { 1 } { 2 } a b ^ { 2 } ) ^ { 6 } + ( - \frac { 18 } { 5 } a ^ { 2 } b ^ { 7 } ) ( + \frac { 10 } { 9 } a ^ { 4 } b ^ { 5 } )$$

Evaluate

$$-\frac{579a^{6}b^{12}}{64}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$324\left(-\frac{1}{4}a^{2}b^{4}\right)^{3}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{7}\times \frac{10}{9}b^{5}$$

To multiply powers of the same base, add their exponents. Add 7 and 5 to get 12.

$$324\left(-\frac{1}{4}a^{2}b^{4}\right)^{3}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Expand \left(-\frac{1}{4}a^{2}b^{4}\right)^{3}.

$$324\left(-\frac{1}{4}\right)^{3}\left(a^{2}\right)^{3}\left(b^{4}\right)^{3}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

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

$$324\left(-\frac{1}{4}\right)^{3}a^{6}\left(b^{4}\right)^{3}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

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

$$324\left(-\frac{1}{4}\right)^{3}a^{6}b^{12}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Calculate -\frac{1}{4} to the power of 3 and get -\frac{1}{64}.

$$324\left(-\frac{1}{64}\right)a^{6}b^{12}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Multiply 324 and -\frac{1}{64} to get -\frac{81}{16}.

$$-\frac{81}{16}a^{6}b^{12}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Expand \left(\frac{1}{2}ab^{2}\right)^{6}.

$$-\frac{81}{16}a^{6}b^{12}+\left(\frac{1}{2}\right)^{6}a^{6}\left(b^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 6 to get 12.

$$-\frac{81}{16}a^{6}b^{12}+\left(\frac{1}{2}\right)^{6}a^{6}b^{12}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Calculate \frac{1}{2} to the power of 6 and get \frac{1}{64}.

$$-\frac{81}{16}a^{6}b^{12}+\frac{1}{64}a^{6}b^{12}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Combine -\frac{81}{16}a^{6}b^{12} and \frac{1}{64}a^{6}b^{12} to get -\frac{323}{64}a^{6}b^{12}.

$$-\frac{323}{64}a^{6}b^{12}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Multiply -\frac{18}{5} and \frac{10}{9} to get -4.

$$-\frac{323}{64}a^{6}b^{12}-4a^{6}b^{12}$$

Combine -\frac{323}{64}a^{6}b^{12} and -4a^{6}b^{12} to get -\frac{579}{64}a^{6}b^{12}.

$$-\frac{579}{64}a^{6}b^{12}$$

Expand

$$-\frac{579a^{6}b^{12}}{64}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$324\left(-\frac{1}{4}a^{2}b^{4}\right)^{3}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{7}\times \frac{10}{9}b^{5}$$

To multiply powers of the same base, add their exponents. Add 7 and 5 to get 12.

$$324\left(-\frac{1}{4}a^{2}b^{4}\right)^{3}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Expand \left(-\frac{1}{4}a^{2}b^{4}\right)^{3}.

$$324\left(-\frac{1}{4}\right)^{3}\left(a^{2}\right)^{3}\left(b^{4}\right)^{3}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

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

$$324\left(-\frac{1}{4}\right)^{3}a^{6}\left(b^{4}\right)^{3}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

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

$$324\left(-\frac{1}{4}\right)^{3}a^{6}b^{12}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Calculate -\frac{1}{4} to the power of 3 and get -\frac{1}{64}.

$$324\left(-\frac{1}{64}\right)a^{6}b^{12}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Multiply 324 and -\frac{1}{64} to get -\frac{81}{16}.

$$-\frac{81}{16}a^{6}b^{12}+\left(\frac{1}{2}ab^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Expand \left(\frac{1}{2}ab^{2}\right)^{6}.

$$-\frac{81}{16}a^{6}b^{12}+\left(\frac{1}{2}\right)^{6}a^{6}\left(b^{2}\right)^{6}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 6 to get 12.

$$-\frac{81}{16}a^{6}b^{12}+\left(\frac{1}{2}\right)^{6}a^{6}b^{12}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Calculate \frac{1}{2} to the power of 6 and get \frac{1}{64}.

$$-\frac{81}{16}a^{6}b^{12}+\frac{1}{64}a^{6}b^{12}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Combine -\frac{81}{16}a^{6}b^{12} and \frac{1}{64}a^{6}b^{12} to get -\frac{323}{64}a^{6}b^{12}.

$$-\frac{323}{64}a^{6}b^{12}-\frac{18}{5}a^{6}b^{12}\times \frac{10}{9}$$

Multiply -\frac{18}{5} and \frac{10}{9} to get -4.

$$-\frac{323}{64}a^{6}b^{12}-4a^{6}b^{12}$$

Combine -\frac{323}{64}a^{6}b^{12} and -4a^{6}b^{12} to get -\frac{579}{64}a^{6}b^{12}.

$$-\frac{579}{64}a^{6}b^{12}$$