Solve (a^4+ib^5)(a^4-ib^5)

$$( a ^ { 4 } + i b ^ { 5 } ) ( a ^ { 4 } - i b ^ { 5 } )$$

Evaluate

$$a^{8}+b^{10}$$

Solution Steps

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

$$\left(a^{4}\right)^{2}-\left(ib^{5}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$a^{8}-\left(ib^{5}\right)^{2}$$

Expand \left(ib^{5}\right)^{2}.

$$a^{8}-i^{2}\left(b^{5}\right)^{2}$$

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

$$a^{8}-i^{2}b^{10}$$

Calculate i to the power of 2 and get -1.

$$a^{8}-\left(-b^{10}\right)$$

The opposite of -b^{10} is b^{10}.

$$a^{8}+b^{10}$$

Expand

$$a^{8}+b^{10}$$

Solution Steps

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

$$\left(a^{4}\right)^{2}-\left(ib^{5}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$a^{8}-\left(ib^{5}\right)^{2}$$

Expand \left(ib^{5}\right)^{2}.

$$a^{8}-i^{2}\left(b^{5}\right)^{2}$$

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

$$a^{8}-i^{2}b^{10}$$

Calculate i to the power of 2 and get -1.

$$a^{8}-\left(-b^{10}\right)$$

The opposite of -b^{10} is b^{10}.

$$a^{8}+b^{10}$$