Solve [(2/5a^{2}-5/2b^{3})(5/2a-2/5b^{2})+frac{1}{100}ab^{2}(16a+625b)](a^3-b^5)+(-b^5)^2

$$[ ( \frac { 2 } { 5 } a ^ { 2 } - \frac { 5 } { 2 } b ^ { 3 } ) ( \frac { 5 } { 2 } a - \frac { 2 } { 5 } b ^ { 2 } ) + \frac { 1 } { 100 } a b ^ { 2 } ( 16 a + 625 b ) ] ( a ^ { 3 } - b ^ { 5 } ) + ( - b ^ { 5 } ) ^ { 2 }$$

Evaluate

$$a^{6}$$

Solution Steps

Use the distributive property to multiply \frac{2}{5}a^{2}-\frac{5}{2}b^{3} by \frac{5}{2}a-\frac{2}{5}b^{2}.

$$\left(a^{3}-\frac{4}{25}a^{2}b^{2}-\frac{25}{4}b^{3}a+b^{5}+\frac{1}{100}ab^{2}\left(16a+625b\right)\right)\left(a^{3}-b^{5}\right)+\left(-b^{5}\right)^{2}$$

Use the distributive property to multiply \frac{1}{100}ab^{2} by 16a+625b.

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

Combine -\frac{4}{25}a^{2}b^{2} and \frac{4}{25}a^{2}b^{2} to get 0.

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

Combine -\frac{25}{4}b^{3}a and \frac{25}{4}ab^{3} to get 0.

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

Consider \left(a^{3}+b^{5}\right)\left(a^{3}-b^{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

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

$$a^{6}-b^{10}+\left(-b^{5}\right)^{2}$$

Calculate -b^{5} to the power of 2 and get \left(b^{5}\right)^{2}.

$$a^{6}-b^{10}+\left(b^{5}\right)^{2}$$

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

$$a^{6}-b^{10}+b^{10}$$

Combine -b^{10} and b^{10} to get 0.

$$a^{6}$$

Expand

$$a^{6}$$

Solution Steps

Use the distributive property to multiply \frac{2}{5}a^{2}-\frac{5}{2}b^{3} by \frac{5}{2}a-\frac{2}{5}b^{2}.

$$\left(a^{3}-\frac{4}{25}a^{2}b^{2}-\frac{25}{4}b^{3}a+b^{5}+\frac{1}{100}ab^{2}\left(16a+625b\right)\right)\left(a^{3}-b^{5}\right)+\left(-b^{5}\right)^{2}$$

Use the distributive property to multiply \frac{1}{100}ab^{2} by 16a+625b.

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

Combine -\frac{4}{25}a^{2}b^{2} and \frac{4}{25}a^{2}b^{2} to get 0.

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

Combine -\frac{25}{4}b^{3}a and \frac{25}{4}ab^{3} to get 0.

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

Consider \left(a^{3}+b^{5}\right)\left(a^{3}-b^{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

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

$$a^{6}-b^{10}+\left(-b^{5}\right)^{2}$$

Calculate -b^{5} to the power of 2 and get \left(b^{5}\right)^{2}.

$$a^{6}-b^{10}+\left(b^{5}\right)^{2}$$

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

$$a^{6}-b^{10}+b^{10}$$

Combine -b^{10} and b^{10} to get 0.

$$a^{6}$$