Solve {[-(4/5)^{2}]^{3}}^{4}:[(4/5)^{6}]^{2}*{[(-3/2)^{2}]^{3}}^2:[(3/5)^2*(-frac{3}{5})^4*(frac{3}{5})^6]

$$\{ [ - ( \frac { 4 } { 5 } ) ^ { 2 } ] ^ { 3 } \} ^ { 4 } : [ ( \frac { 4 } { 5 } ) ^ { 6 } ] ^ { 2 } \cdot \{ [ ( - \frac { 3 } { 2 } ) ^ { 2 } ] ^ { 3 } \} ^ { 2 } : [ ( \frac { 3 } { 5 } ) ^ { 2 } \cdot ( - \frac { 3 } { 5 } ) ^ { 4 } \cdot ( \frac { 3 } { 5 } ) ^ { 6 } ]$$

Evaluate

$$4096$$

Solution Steps

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

$$\frac{\frac{\left(-\left(\frac{4}{5}\right)^{2}\right)^{12}}{\left(\left(\frac{4}{5}\right)^{6}\right)^{2}}\left(\left(\left(-\frac{3}{2}\right)^{2}\right)^{3}\right)^{2}}{\left(\frac{3}{5}\right)^{2}\left(-\frac{3}{5}\right)^{4}\times \left(\frac{3}{5}\right)^{6}}$$

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

$$\frac{\frac{\left(-\left(\frac{4}{5}\right)^{2}\right)^{12}}{\left(\frac{4}{5}\right)^{12}}\left(\left(\left(-\frac{3}{2}\right)^{2}\right)^{3}\right)^{2}}{\left(\frac{3}{5}\right)^{2}\left(-\frac{3}{5}\right)^{4}\times \left(\frac{3}{5}\right)^{6}}$$

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

$$\frac{\frac{\left(-\left(\frac{4}{5}\right)^{2}\right)^{12}}{\left(\frac{4}{5}\right)^{12}}\left(\left(-\frac{3}{2}\right)^{6}\right)^{2}}{\left(\frac{3}{5}\right)^{2}\left(-\frac{3}{5}\right)^{4}\times \left(\frac{3}{5}\right)^{6}}$$

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

$$\frac{\frac{\left(-\left(\frac{4}{5}\right)^{2}\right)^{12}}{\left(\frac{4}{5}\right)^{12}}\left(-\frac{3}{2}\right)^{12}}{\left(\frac{3}{5}\right)^{2}\left(-\frac{3}{5}\right)^{4}\times \left(\frac{3}{5}\right)^{6}}$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

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

Calculate \frac{4}{5} to the power of 2 and get \frac{16}{25}.

$$\frac{\frac{\left(-\frac{16}{25}\right)^{12}}{\left(\frac{4}{5}\right)^{12}}\left(-\frac{3}{2}\right)^{12}}{\left(\frac{3}{5}\right)^{8}\left(-\frac{3}{5}\right)^{4}}$$

Calculate -\frac{16}{25} to the power of 12 and get \frac{281474976710656}{59604644775390625}.

$$\frac{\frac{\frac{281474976710656}{59604644775390625}}{\left(\frac{4}{5}\right)^{12}}\left(-\frac{3}{2}\right)^{12}}{\left(\frac{3}{5}\right)^{8}\left(-\frac{3}{5}\right)^{4}}$$

Calculate \frac{4}{5} to the power of 12 and get \frac{16777216}{244140625}.

$$\frac{\frac{\frac{281474976710656}{59604644775390625}}{\frac{16777216}{244140625}}\left(-\frac{3}{2}\right)^{12}}{\left(\frac{3}{5}\right)^{8}\left(-\frac{3}{5}\right)^{4}}$$

Divide \frac{281474976710656}{59604644775390625} by \frac{16777216}{244140625} by multiplying \frac{281474976710656}{59604644775390625} by the reciprocal of \frac{16777216}{244140625}.

$$\frac{\frac{281474976710656}{59604644775390625}\times \frac{244140625}{16777216}\left(-\frac{3}{2}\right)^{12}}{\left(\frac{3}{5}\right)^{8}\left(-\frac{3}{5}\right)^{4}}$$

Multiply \frac{281474976710656}{59604644775390625} and \frac{244140625}{16777216} to get \frac{16777216}{244140625}.

$$\frac{\frac{16777216}{244140625}\left(-\frac{3}{2}\right)^{12}}{\left(\frac{3}{5}\right)^{8}\left(-\frac{3}{5}\right)^{4}}$$

Calculate -\frac{3}{2} to the power of 12 and get \frac{531441}{4096}.

$$\frac{\frac{16777216}{244140625}\times \frac{531441}{4096}}{\left(\frac{3}{5}\right)^{8}\left(-\frac{3}{5}\right)^{4}}$$

Multiply \frac{16777216}{244140625} and \frac{531441}{4096} to get \frac{2176782336}{244140625}.

$$\frac{\frac{2176782336}{244140625}}{\left(\frac{3}{5}\right)^{8}\left(-\frac{3}{5}\right)^{4}}$$

Calculate \frac{3}{5} to the power of 8 and get \frac{6561}{390625}.

$$\frac{\frac{2176782336}{244140625}}{\frac{6561}{390625}\left(-\frac{3}{5}\right)^{4}}$$

Calculate -\frac{3}{5} to the power of 4 and get \frac{81}{625}.

$$\frac{\frac{2176782336}{244140625}}{\frac{6561}{390625}\times \frac{81}{625}}$$

Multiply \frac{6561}{390625} and \frac{81}{625} to get \frac{531441}{244140625}.

$$\frac{\frac{2176782336}{244140625}}{\frac{531441}{244140625}}$$

Divide \frac{2176782336}{244140625} by \frac{531441}{244140625} by multiplying \frac{2176782336}{244140625} by the reciprocal of \frac{531441}{244140625}.

$$\frac{2176782336}{244140625}\times \frac{244140625}{531441}$$

Multiply \frac{2176782336}{244140625} and \frac{244140625}{531441} to get 4096.

$$4096$$

Factor

$$2^{12}$$