Solve (1+omega)^3-(1+omega^2)^3=?

$$( 1 + \omega ) ^ { 3 } - ( 1 + \omega ^ { 2 } ) ^ { 3 } = ?$$

Evaluate

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

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(1+\omega \right)^{3}.

$$1+3\omega +3\omega ^{2}+\omega ^{3}-\left(1+\omega ^{2}\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(1+\omega ^{2}\right)^{3}.

$$1+3\omega +3\omega ^{2}+\omega ^{3}-\left(1+3\omega ^{2}+3\left(\omega ^{2}\right)^{2}+\left(\omega ^{2}\right)^{3}\right)$$

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

$$1+3\omega +3\omega ^{2}+\omega ^{3}-\left(1+3\omega ^{2}+3\omega ^{4}+\left(\omega ^{2}\right)^{3}\right)$$

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

$$1+3\omega +3\omega ^{2}+\omega ^{3}-\left(1+3\omega ^{2}+3\omega ^{4}+\omega ^{6}\right)$$

To find the opposite of 1+3\omega ^{2}+3\omega ^{4}+\omega ^{6}, find the opposite of each term.

$$1+3\omega +3\omega ^{2}+\omega ^{3}-1-3\omega ^{2}-3\omega ^{4}-\omega ^{6}$$

Subtract 1 from 1 to get 0.

$$3\omega +3\omega ^{2}+\omega ^{3}-3\omega ^{2}-3\omega ^{4}-\omega ^{6}$$

Combine 3\omega ^{2} and -3\omega ^{2} to get 0.

$$3\omega +\omega ^{3}-3\omega ^{4}-\omega ^{6}$$

Expand

$$3\omega +\omega ^{3}-3\omega ^{4}-\omega ^{6}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(1+\omega \right)^{3}.

$$1+3\omega +3\omega ^{2}+\omega ^{3}-\left(1+\omega ^{2}\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(1+\omega ^{2}\right)^{3}.

$$1+3\omega +3\omega ^{2}+\omega ^{3}-\left(1+3\omega ^{2}+3\left(\omega ^{2}\right)^{2}+\left(\omega ^{2}\right)^{3}\right)$$

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

$$1+3\omega +3\omega ^{2}+\omega ^{3}-\left(1+3\omega ^{2}+3\omega ^{4}+\left(\omega ^{2}\right)^{3}\right)$$

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

$$1+3\omega +3\omega ^{2}+\omega ^{3}-\left(1+3\omega ^{2}+3\omega ^{4}+\omega ^{6}\right)$$

To find the opposite of 1+3\omega ^{2}+3\omega ^{4}+\omega ^{6}, find the opposite of each term.

$$1+3\omega +3\omega ^{2}+\omega ^{3}-1-3\omega ^{2}-3\omega ^{4}-\omega ^{6}$$

Subtract 1 from 1 to get 0.

$$3\omega +3\omega ^{2}+\omega ^{3}-3\omega ^{2}-3\omega ^{4}-\omega ^{6}$$

Combine 3\omega ^{2} and -3\omega ^{2} to get 0.

$$3\omega +\omega ^{3}-3\omega ^{4}-\omega ^{6}$$