Solve (8)(3+sqrt{2})^2(3-sqrt{2})-(3-sqrt{2})^2(3+sqrt{2})

$$( 8 ) ( 3 + \sqrt { 2 } ) ^ { 2 } ( 3 - \sqrt { 2 } ) - ( 3 - \sqrt { 2 } ) ^ { 2 } ( 3 + \sqrt { 2 } )$$

Evaluate

$$63\sqrt{2}+147\approx 236.09545443$$

Solution Steps

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

$$8\left(9+6\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)\left(3-\sqrt{2}\right)-\left(3-\sqrt{2}\right)^{2}\left(3+\sqrt{2}\right)$$

The square of \sqrt{2} is 2.

$$8\left(9+6\sqrt{2}+2\right)\left(3-\sqrt{2}\right)-\left(3-\sqrt{2}\right)^{2}\left(3+\sqrt{2}\right)$$

Add 9 and 2 to get 11.

$$8\left(11+6\sqrt{2}\right)\left(3-\sqrt{2}\right)-\left(3-\sqrt{2}\right)^{2}\left(3+\sqrt{2}\right)$$

Use the distributive property to multiply 8 by 11+6\sqrt{2}.

$$\left(88+48\sqrt{2}\right)\left(3-\sqrt{2}\right)-\left(3-\sqrt{2}\right)^{2}\left(3+\sqrt{2}\right)$$

Use the distributive property to multiply 88+48\sqrt{2} by 3-\sqrt{2} and combine like terms.

$$264+56\sqrt{2}-48\left(\sqrt{2}\right)^{2}-\left(3-\sqrt{2}\right)^{2}\left(3+\sqrt{2}\right)$$

The square of \sqrt{2} is 2.

$$264+56\sqrt{2}-48\times 2-\left(3-\sqrt{2}\right)^{2}\left(3+\sqrt{2}\right)$$

Multiply -48 and 2 to get -96.

$$264+56\sqrt{2}-96-\left(3-\sqrt{2}\right)^{2}\left(3+\sqrt{2}\right)$$

Subtract 96 from 264 to get 168.

$$168+56\sqrt{2}-\left(3-\sqrt{2}\right)^{2}\left(3+\sqrt{2}\right)$$

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

$$168+56\sqrt{2}-\left(9-6\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)\left(3+\sqrt{2}\right)$$

The square of \sqrt{2} is 2.

$$168+56\sqrt{2}-\left(9-6\sqrt{2}+2\right)\left(3+\sqrt{2}\right)$$

Add 9 and 2 to get 11.

$$168+56\sqrt{2}-\left(11-6\sqrt{2}\right)\left(3+\sqrt{2}\right)$$

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

$$168+56\sqrt{2}-\left(33-7\sqrt{2}-6\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$168+56\sqrt{2}-\left(33-7\sqrt{2}-6\times 2\right)$$

Multiply -6 and 2 to get -12.

$$168+56\sqrt{2}-\left(33-7\sqrt{2}-12\right)$$

Subtract 12 from 33 to get 21.

$$168+56\sqrt{2}-\left(21-7\sqrt{2}\right)$$

To find the opposite of 21-7\sqrt{2}, find the opposite of each term.

$$168+56\sqrt{2}-21+7\sqrt{2}$$

Subtract 21 from 168 to get 147.

$$147+56\sqrt{2}+7\sqrt{2}$$

Combine 56\sqrt{2} and 7\sqrt{2} to get 63\sqrt{2}.

$$147+63\sqrt{2}$$

Factor

$$21 {(3 \sqrt{2} + 7)} = 236.09545443$$