Solve (3-sqrt{2})^2(3+sqrt{2})+(3+sqrt{2})^2(3-sqrt{2})

$$( 3 - \sqrt { 2 } ) ^ { 2 } ( 3 + \sqrt { 2 } ) + ( 3 + \sqrt { 2 } ) ^ { 2 } ( 3 - \sqrt { 2 } )$$

Evaluate

$$42$$

Solution Steps

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

$$\left(9-6\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)\left(3+\sqrt{2}\right)+\left(3+\sqrt{2}\right)^{2}\left(3-\sqrt{2}\right)$$

The square of \sqrt{2} is 2.

$$\left(9-6\sqrt{2}+2\right)\left(3+\sqrt{2}\right)+\left(3+\sqrt{2}\right)^{2}\left(3-\sqrt{2}\right)$$

Add 9 and 2 to get 11.

$$\left(11-6\sqrt{2}\right)\left(3+\sqrt{2}\right)+\left(3+\sqrt{2}\right)^{2}\left(3-\sqrt{2}\right)$$

Use the distributive property to multiply 11-6\sqrt{2} by 3+\sqrt{2} and combine like terms.

$$33-7\sqrt{2}-6\left(\sqrt{2}\right)^{2}+\left(3+\sqrt{2}\right)^{2}\left(3-\sqrt{2}\right)$$

The square of \sqrt{2} is 2.

$$33-7\sqrt{2}-6\times 2+\left(3+\sqrt{2}\right)^{2}\left(3-\sqrt{2}\right)$$

Multiply -6 and 2 to get -12.

$$33-7\sqrt{2}-12+\left(3+\sqrt{2}\right)^{2}\left(3-\sqrt{2}\right)$$

Subtract 12 from 33 to get 21.

$$21-7\sqrt{2}+\left(3+\sqrt{2}\right)^{2}\left(3-\sqrt{2}\right)$$

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

$$21-7\sqrt{2}+\left(9+6\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)\left(3-\sqrt{2}\right)$$

The square of \sqrt{2} is 2.

$$21-7\sqrt{2}+\left(9+6\sqrt{2}+2\right)\left(3-\sqrt{2}\right)$$

Add 9 and 2 to get 11.

$$21-7\sqrt{2}+\left(11+6\sqrt{2}\right)\left(3-\sqrt{2}\right)$$

Use the distributive property to multiply 11+6\sqrt{2} by 3-\sqrt{2} and combine like terms.

$$21-7\sqrt{2}+33+7\sqrt{2}-6\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$21-7\sqrt{2}+33+7\sqrt{2}-6\times 2$$

Multiply -6 and 2 to get -12.

$$21-7\sqrt{2}+33+7\sqrt{2}-12$$

Subtract 12 from 33 to get 21.

$$21-7\sqrt{2}+21+7\sqrt{2}$$

Add 21 and 21 to get 42.

$$42-7\sqrt{2}+7\sqrt{2}$$

Combine -7\sqrt{2} and 7\sqrt{2} to get 0.

$$42$$

Factor

$$2\times 3\times 7$$