Solve [(-4/9a^{3}b^{4})^{6}:(-2/3ab^{2})^{6}(2/3a^{2}b^{2})^{2}]:[(-4/15ab^{2})^{5}(-frac{5}{2}a)^{5}:(frac{2}{3}a^2b^2)^3]^7

$$[ ( - \frac { 4 } { 9 } a ^ { 3 } b ^ { 4 } ) ^ { 6 } : ( - \frac { 2 } { 3 } a b ^ { 2 } ) ^ { 6 } ( \frac { 2 } { 3 } a ^ { 2 } b ^ { 2 } ) ^ { 2 } ] : [ ( - \frac { 4 } { 15 } a b ^ { 2 } ) ^ { 5 } ( - \frac { 5 } { 2 } a ) ^ { 5 } : ( \frac { 2 } { 3 } a ^ { 2 } b ^ { 2 } ) ^ { 3 } ] ^ { 7 }$$

Evaluate

$$\frac{729}{64\left(ab\right)^{12}}$$

Solution Steps

Expand \left(-\frac{4}{9}a^{3}b^{4}\right)^{6}.

$$\frac{\frac{\left(-\frac{4}{9}\right)^{6}\left(a^{3}\right)^{6}\left(b^{4}\right)^{6}}{\left(-\frac{2}{3}ab^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 6 to get 18.

$$\frac{\frac{\left(-\frac{4}{9}\right)^{6}a^{18}\left(b^{4}\right)^{6}}{\left(-\frac{2}{3}ab^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 6 to get 24.

$$\frac{\frac{\left(-\frac{4}{9}\right)^{6}a^{18}b^{24}}{\left(-\frac{2}{3}ab^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate -\frac{4}{9} to the power of 6 and get \frac{4096}{531441}.

$$\frac{\frac{\frac{4096}{531441}a^{18}b^{24}}{\left(-\frac{2}{3}ab^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Expand \left(-\frac{2}{3}ab^{2}\right)^{6}.

$$\frac{\frac{\frac{4096}{531441}a^{18}b^{24}}{\left(-\frac{2}{3}\right)^{6}a^{6}\left(b^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{\frac{4096}{531441}a^{18}b^{24}}{\left(-\frac{2}{3}\right)^{6}a^{6}b^{12}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate -\frac{2}{3} to the power of 6 and get \frac{64}{729}.

$$\frac{\frac{\frac{4096}{531441}a^{18}b^{24}}{\frac{64}{729}a^{6}b^{12}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Cancel out a^{6}b^{12} in both numerator and denominator.

$$\frac{\frac{\frac{4096}{531441}a^{12}b^{12}}{\frac{64}{729}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Divide \frac{4096}{531441}a^{12}b^{12} by \frac{64}{729} by multiplying \frac{4096}{531441}a^{12}b^{12} by the reciprocal of \frac{64}{729}.

$$\frac{\frac{\frac{4096}{531441}a^{12}b^{12}\times 729}{64}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Multiply \frac{4096}{531441} and 729 to get \frac{4096}{729}.

$$\frac{\frac{\frac{4096}{729}a^{12}b^{12}}{64}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Divide \frac{4096}{729}a^{12}b^{12} by 64 to get \frac{64}{729}a^{12}b^{12}.

$$\frac{\frac{64}{729}a^{12}b^{12}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{64}{729}a^{12}b^{12}\times \left(\frac{2}{3}\right)^{2}\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{64}{729}a^{12}b^{12}\times \left(\frac{2}{3}\right)^{2}a^{4}\left(b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{64}{729}a^{12}b^{12}\times \left(\frac{2}{3}\right)^{2}a^{4}b^{4}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{64}{729}a^{12}b^{12}\times \frac{4}{9}a^{4}b^{4}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Multiply \frac{64}{729} and \frac{4}{9} to get \frac{256}{6561}.

$$\frac{\frac{256}{6561}a^{12}b^{12}a^{4}b^{4}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To multiply powers of the same base, add their exponents. Add 12 and 4 to get 16.

$$\frac{\frac{256}{6561}a^{16}b^{12}b^{4}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To multiply powers of the same base, add their exponents. Add 12 and 4 to get 16.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Expand \left(-\frac{4}{15}ab^{2}\right)^{5}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\left(-\frac{4}{15}\right)^{5}a^{5}\left(b^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\left(-\frac{4}{15}\right)^{5}a^{5}b^{10}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate -\frac{4}{15} to the power of 5 and get -\frac{1024}{759375}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{-\frac{1024}{759375}a^{5}b^{10}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Expand \left(-\frac{5}{2}a\right)^{5}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{-\frac{1024}{759375}a^{5}b^{10}\left(-\frac{5}{2}\right)^{5}a^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate -\frac{5}{2} to the power of 5 and get -\frac{3125}{32}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{-\frac{1024}{759375}a^{5}b^{10}\left(-\frac{3125}{32}\right)a^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Multiply -\frac{1024}{759375} and -\frac{3125}{32} to get \frac{32}{243}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{5}b^{10}a^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To multiply powers of the same base, add their exponents. Add 5 and 5 to get 10.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\left(\frac{2}{3}\right)^{3}\left(a^{2}\right)^{3}\left(b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\left(\frac{2}{3}\right)^{3}a^{6}\left(b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\left(\frac{2}{3}\right)^{3}a^{6}b^{6}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\frac{8}{27}a^{6}b^{6}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{4}b^{4}}{\frac{8}{27}}\right)^{7}}$$

Divide \frac{32}{243}a^{4}b^{4} by \frac{8}{27} by multiplying \frac{32}{243}a^{4}b^{4} by the reciprocal of \frac{8}{27}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{4}b^{4}\times 27}{8}\right)^{7}}$$

Multiply \frac{32}{243} and 27 to get \frac{32}{9}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{9}a^{4}b^{4}}{8}\right)^{7}}$$

Divide \frac{32}{9}a^{4}b^{4} by 8 to get \frac{4}{9}a^{4}b^{4}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{4}{9}a^{4}b^{4}\right)^{7}}$$

Expand \left(\frac{4}{9}a^{4}b^{4}\right)^{7}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{4}{9}\right)^{7}\left(a^{4}\right)^{7}\left(b^{4}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 7 to get 28.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{4}{9}\right)^{7}a^{28}\left(b^{4}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 7 to get 28.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{4}{9}\right)^{7}a^{28}b^{28}}$$

Calculate \frac{4}{9} to the power of 7 and get \frac{16384}{4782969}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\frac{16384}{4782969}a^{28}b^{28}}$$

Cancel out a^{16}b^{16} in both numerator and denominator.

$$\frac{\frac{256}{6561}}{\frac{16384}{4782969}a^{12}b^{12}}$$

Express \frac{\frac{256}{6561}}{\frac{16384}{4782969}a^{12}b^{12}} as a single fraction.

$$\frac{256}{6561\times \frac{16384}{4782969}a^{12}b^{12}}$$

Multiply 6561 and \frac{16384}{4782969} to get \frac{16384}{729}.

$$\frac{256}{\frac{16384}{729}a^{12}b^{12}}$$

Expand

$$\frac{729}{64\left(ab\right)^{12}}$$

Solution Steps

Expand \left(-\frac{4}{9}a^{3}b^{4}\right)^{6}.

$$\frac{\frac{\left(-\frac{4}{9}\right)^{6}\left(a^{3}\right)^{6}\left(b^{4}\right)^{6}}{\left(-\frac{2}{3}ab^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 6 to get 18.

$$\frac{\frac{\left(-\frac{4}{9}\right)^{6}a^{18}\left(b^{4}\right)^{6}}{\left(-\frac{2}{3}ab^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 6 to get 24.

$$\frac{\frac{\left(-\frac{4}{9}\right)^{6}a^{18}b^{24}}{\left(-\frac{2}{3}ab^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate -\frac{4}{9} to the power of 6 and get \frac{4096}{531441}.

$$\frac{\frac{\frac{4096}{531441}a^{18}b^{24}}{\left(-\frac{2}{3}ab^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Expand \left(-\frac{2}{3}ab^{2}\right)^{6}.

$$\frac{\frac{\frac{4096}{531441}a^{18}b^{24}}{\left(-\frac{2}{3}\right)^{6}a^{6}\left(b^{2}\right)^{6}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{\frac{4096}{531441}a^{18}b^{24}}{\left(-\frac{2}{3}\right)^{6}a^{6}b^{12}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate -\frac{2}{3} to the power of 6 and get \frac{64}{729}.

$$\frac{\frac{\frac{4096}{531441}a^{18}b^{24}}{\frac{64}{729}a^{6}b^{12}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Cancel out a^{6}b^{12} in both numerator and denominator.

$$\frac{\frac{\frac{4096}{531441}a^{12}b^{12}}{\frac{64}{729}}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Divide \frac{4096}{531441}a^{12}b^{12} by \frac{64}{729} by multiplying \frac{4096}{531441}a^{12}b^{12} by the reciprocal of \frac{64}{729}.

$$\frac{\frac{\frac{4096}{531441}a^{12}b^{12}\times 729}{64}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Multiply \frac{4096}{531441} and 729 to get \frac{4096}{729}.

$$\frac{\frac{\frac{4096}{729}a^{12}b^{12}}{64}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Divide \frac{4096}{729}a^{12}b^{12} by 64 to get \frac{64}{729}a^{12}b^{12}.

$$\frac{\frac{64}{729}a^{12}b^{12}\times \left(\frac{2}{3}a^{2}b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{64}{729}a^{12}b^{12}\times \left(\frac{2}{3}\right)^{2}\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{64}{729}a^{12}b^{12}\times \left(\frac{2}{3}\right)^{2}a^{4}\left(b^{2}\right)^{2}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{64}{729}a^{12}b^{12}\times \left(\frac{2}{3}\right)^{2}a^{4}b^{4}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{64}{729}a^{12}b^{12}\times \frac{4}{9}a^{4}b^{4}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Multiply \frac{64}{729} and \frac{4}{9} to get \frac{256}{6561}.

$$\frac{\frac{256}{6561}a^{12}b^{12}a^{4}b^{4}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To multiply powers of the same base, add their exponents. Add 12 and 4 to get 16.

$$\frac{\frac{256}{6561}a^{16}b^{12}b^{4}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To multiply powers of the same base, add their exponents. Add 12 and 4 to get 16.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\left(-\frac{4}{15}ab^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Expand \left(-\frac{4}{15}ab^{2}\right)^{5}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\left(-\frac{4}{15}\right)^{5}a^{5}\left(b^{2}\right)^{5}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\left(-\frac{4}{15}\right)^{5}a^{5}b^{10}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate -\frac{4}{15} to the power of 5 and get -\frac{1024}{759375}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{-\frac{1024}{759375}a^{5}b^{10}\left(-\frac{5}{2}a\right)^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Expand \left(-\frac{5}{2}a\right)^{5}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{-\frac{1024}{759375}a^{5}b^{10}\left(-\frac{5}{2}\right)^{5}a^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Calculate -\frac{5}{2} to the power of 5 and get -\frac{3125}{32}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{-\frac{1024}{759375}a^{5}b^{10}\left(-\frac{3125}{32}\right)a^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

Multiply -\frac{1024}{759375} and -\frac{3125}{32} to get \frac{32}{243}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{5}b^{10}a^{5}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

To multiply powers of the same base, add their exponents. Add 5 and 5 to get 10.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\left(\frac{2}{3}a^{2}b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\left(\frac{2}{3}\right)^{3}\left(a^{2}\right)^{3}\left(b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\left(\frac{2}{3}\right)^{3}a^{6}\left(b^{2}\right)^{3}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\left(\frac{2}{3}\right)^{3}a^{6}b^{6}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{10}b^{10}}{\frac{8}{27}a^{6}b^{6}}\right)^{7}}$$

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

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{4}b^{4}}{\frac{8}{27}}\right)^{7}}$$

Divide \frac{32}{243}a^{4}b^{4} by \frac{8}{27} by multiplying \frac{32}{243}a^{4}b^{4} by the reciprocal of \frac{8}{27}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{243}a^{4}b^{4}\times 27}{8}\right)^{7}}$$

Multiply \frac{32}{243} and 27 to get \frac{32}{9}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{\frac{32}{9}a^{4}b^{4}}{8}\right)^{7}}$$

Divide \frac{32}{9}a^{4}b^{4} by 8 to get \frac{4}{9}a^{4}b^{4}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{4}{9}a^{4}b^{4}\right)^{7}}$$

Expand \left(\frac{4}{9}a^{4}b^{4}\right)^{7}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{4}{9}\right)^{7}\left(a^{4}\right)^{7}\left(b^{4}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 7 to get 28.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{4}{9}\right)^{7}a^{28}\left(b^{4}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 7 to get 28.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\left(\frac{4}{9}\right)^{7}a^{28}b^{28}}$$

Calculate \frac{4}{9} to the power of 7 and get \frac{16384}{4782969}.

$$\frac{\frac{256}{6561}a^{16}b^{16}}{\frac{16384}{4782969}a^{28}b^{28}}$$

Cancel out a^{16}b^{16} in both numerator and denominator.

$$\frac{\frac{256}{6561}}{\frac{16384}{4782969}a^{12}b^{12}}$$

Express \frac{\frac{256}{6561}}{\frac{16384}{4782969}a^{12}b^{12}} as a single fraction.

$$\frac{256}{6561\times \frac{16384}{4782969}a^{12}b^{12}}$$

Multiply 6561 and \frac{16384}{4782969} to get \frac{16384}{729}.

$$\frac{256}{\frac{16384}{729}a^{12}b^{12}}$$