Solve 25(-1/5a^{2}b)^{3}+(a^{3}b+4a^{3}b)^2*(-1/5b)+1/3a^2b(a^2b)^2

$$25 ( - \frac { 1 } { 5 } a ^ { 2 } b ) ^ { 3 } + ( a ^ { 3 } b + 4 a ^ { 3 } b ) ^ { 2 } \cdot ( - \frac { 1 } { 5 } b ) + \frac { 1 } { 3 } a ^ { 2 } b ( a ^ { 2 } b ) ^ { 2 }$$

Evaluate

$$-\frac{73b^{3}a^{6}}{15}$$

Solution Steps

Expand \left(-\frac{1}{5}a^{2}b\right)^{3}.

$$25\left(-\frac{1}{5}\right)^{3}\left(a^{2}\right)^{3}b^{3}+\left(a^{3}b+4a^{3}b\right)^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

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

Calculate -\frac{1}{5} to the power of 3 and get -\frac{1}{125}.

$$25\left(-\frac{1}{125}\right)a^{6}b^{3}+\left(a^{3}b+4a^{3}b\right)^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

Multiply 25 and -\frac{1}{125} to get -\frac{1}{5}.

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

Combine a^{3}b and 4a^{3}b to get 5a^{3}b.

$$-\frac{1}{5}a^{6}b^{3}+\left(5a^{3}b\right)^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

$$-\frac{1}{5}a^{6}b^{3}+5^{2}\left(a^{3}\right)^{2}b^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

$$-\frac{1}{5}a^{6}b^{3}+5^{2}a^{6}b^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$-\frac{1}{5}a^{6}b^{3}+25a^{6}b^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

Multiply 25 and -\frac{1}{5} to get -5.

$$-\frac{1}{5}a^{6}b^{3}-5a^{6}b^{2}b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

$$-\frac{1}{5}a^{6}b^{3}-5a^{6}b^{3}+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

Combine -\frac{1}{5}a^{6}b^{3} and -5a^{6}b^{3} to get -\frac{26}{5}a^{6}b^{3}.

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{2}b\left(a^{2}\right)^{2}b^{2}$$

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

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{2}ba^{4}b^{2}$$

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

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{6}bb^{2}$$

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

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{6}b^{3}$$

Combine -\frac{26}{5}a^{6}b^{3} and \frac{1}{3}a^{6}b^{3} to get -\frac{73}{15}a^{6}b^{3}.

$$-\frac{73}{15}a^{6}b^{3}$$

Expand

$$-\frac{73b^{3}a^{6}}{15}$$

Solution Steps

Expand \left(-\frac{1}{5}a^{2}b\right)^{3}.

$$25\left(-\frac{1}{5}\right)^{3}\left(a^{2}\right)^{3}b^{3}+\left(a^{3}b+4a^{3}b\right)^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

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

Calculate -\frac{1}{5} to the power of 3 and get -\frac{1}{125}.

$$25\left(-\frac{1}{125}\right)a^{6}b^{3}+\left(a^{3}b+4a^{3}b\right)^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

Multiply 25 and -\frac{1}{125} to get -\frac{1}{5}.

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

Combine a^{3}b and 4a^{3}b to get 5a^{3}b.

$$-\frac{1}{5}a^{6}b^{3}+\left(5a^{3}b\right)^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

$$-\frac{1}{5}a^{6}b^{3}+5^{2}\left(a^{3}\right)^{2}b^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

$$-\frac{1}{5}a^{6}b^{3}+5^{2}a^{6}b^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$-\frac{1}{5}a^{6}b^{3}+25a^{6}b^{2}\left(-\frac{1}{5}\right)b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

Multiply 25 and -\frac{1}{5} to get -5.

$$-\frac{1}{5}a^{6}b^{3}-5a^{6}b^{2}b+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

$$-\frac{1}{5}a^{6}b^{3}-5a^{6}b^{3}+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

Combine -\frac{1}{5}a^{6}b^{3} and -5a^{6}b^{3} to get -\frac{26}{5}a^{6}b^{3}.

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{2}b\left(a^{2}b\right)^{2}$$

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

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{2}b\left(a^{2}\right)^{2}b^{2}$$

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

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{2}ba^{4}b^{2}$$

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

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{6}bb^{2}$$

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

$$-\frac{26}{5}a^{6}b^{3}+\frac{1}{3}a^{6}b^{3}$$

Combine -\frac{26}{5}a^{6}b^{3} and \frac{1}{3}a^{6}b^{3} to get -\frac{73}{15}a^{6}b^{3}.

$$-\frac{73}{15}a^{6}b^{3}$$