Solve [(-9/5ab)*(-5/3a)^{2}]^{2}-4a^{4}[2/3a^{2}b*(-3/4b)]^{1}-5a^{4}[(-frac{1}{2}ab)^2]^0*(-a^2b^2)

$$[ ( - \frac { 9 } { 5 } a b ) \cdot ( - \frac { 5 } { 3 } a ) ^ { 2 } ] ^ { 2 } - 4 a ^ { 4 } [ \frac { 2 } { 3 } a ^ { 2 } b \cdot ( - \frac { 3 } { 4 } b ) ] ^ { 1 } - 5 a ^ { 4 } [ ( - \frac { 1 } { 2 } a b ) ^ { 2 } ] ^ { 0 } \cdot ( - a ^ { 2 } b ^ { 2 } )$$

Evaluate

$$32b^{2}a^{6}$$

Solution Steps

Multiply b and b to get b^{2}.

$$\left(-\frac{9}{5}ab\left(-\frac{5}{3}a\right)^{2}\right)^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(\left(-\frac{1}{2}ab\right)^{2}\right)^{0}\left(-a^{2}\right)b^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 0 to get 0.

$$\left(-\frac{9}{5}ab\left(-\frac{5}{3}a\right)^{2}\right)^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Expand \left(-\frac{5}{3}a\right)^{2}.

$$\left(-\frac{9}{5}ab\left(-\frac{5}{3}\right)^{2}a^{2}\right)^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Calculate -\frac{5}{3} to the power of 2 and get \frac{25}{9}.

$$\left(-\frac{9}{5}ab\times \frac{25}{9}a^{2}\right)^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Multiply -\frac{9}{5} and \frac{25}{9} to get -5.

$$\left(-5aba^{2}\right)^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(-5a^{3}b\right)^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Expand \left(-5a^{3}b\right)^{2}.

$$\left(-5\right)^{2}\left(a^{3}\right)^{2}b^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

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

$$\left(-5\right)^{2}a^{6}b^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Calculate -5 to the power of 2 and get 25.

$$25a^{6}b^{2}-4a^{4}\times \left(\frac{2}{3}a^{2}b^{2}\left(-\frac{3}{4}\right)\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Multiply \frac{2}{3} and -\frac{3}{4} to get -\frac{1}{2}.

$$25a^{6}b^{2}-4a^{4}\left(-\frac{1}{2}a^{2}b^{2}\right)^{1}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Calculate -\frac{1}{2}a^{2}b^{2} to the power of 1 and get -\frac{1}{2}a^{2}b^{2}.

$$25a^{6}b^{2}-4a^{4}\left(-\frac{1}{2}\right)a^{2}b^{2}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Multiply 4 and -\frac{1}{2} to get -2.

$$25a^{6}b^{2}-\left(-2a^{4}a^{2}b^{2}\right)-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

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

$$25a^{6}b^{2}-\left(-2a^{6}b^{2}\right)-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

The opposite of -2a^{6}b^{2} is 2a^{6}b^{2}.

$$25a^{6}b^{2}+2a^{6}b^{2}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Combine 25a^{6}b^{2} and 2a^{6}b^{2} to get 27a^{6}b^{2}.

$$27a^{6}b^{2}-5a^{4}\left(-\frac{1}{2}ab\right)^{0}\left(-a^{2}\right)b^{2}$$

Calculate -\frac{1}{2}ab to the power of 0 and get 1.

$$27a^{6}b^{2}-5a^{4}\times 1\left(-a^{2}\right)b^{2}$$

Multiply 5 and 1 to get 5.

$$27a^{6}b^{2}-5a^{4}\left(-a^{2}\right)b^{2}$$

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

$$27a^{6}b^{2}-5a^{6}\left(-1\right)b^{2}$$

Multiply -5 and -1 to get 5.

$$27a^{6}b^{2}+5a^{6}b^{2}$$

Combine 27a^{6}b^{2} and 5a^{6}b^{2} to get 32a^{6}b^{2}.

$$32a^{6}b^{2}$$

Differentiate w.r.t. a

$$192b^{2}a^{5}$$