Solve =(3sqrt{2}-2sqrt{3})(3sqrt{2}+2sqrt{3})-sqrt{72}-(3-sqrt{2})^2

$$= ( 3 \sqrt { 2 } - 2 \sqrt { 3 } ) ( 3 \sqrt { 2 } + 2 \sqrt { 3 } ) - \sqrt { 72 } - ( 3 - \sqrt { 2 } ) ^ { 2 }$$

Evaluate

$$-5$$

Solution Steps

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

$$\left(3\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

Expand \left(3\sqrt{2}\right)^{2}.

$$3^{2}\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$9\times 2-\left(2\sqrt{3}\right)^{2}-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

Multiply 9 and 2 to get 18.

$$18-\left(2\sqrt{3}\right)^{2}-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$18-2^{2}\left(\sqrt{3}\right)^{2}-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$18-4\left(\sqrt{3}\right)^{2}-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$18-4\times 3-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$18-12-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

Subtract 12 from 18 to get 6.

$$6-\sqrt{72}-\left(3-\sqrt{2}\right)^{2}$$

Factor 72=6^{2}\times 2. Rewrite the square root of the product \sqrt{6^{2}\times 2} as the product of square roots \sqrt{6^{2}}\sqrt{2}. Take the square root of 6^{2}.

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

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

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

The square of \sqrt{2} is 2.

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

Add 9 and 2 to get 11.

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

To find the opposite of 11-6\sqrt{2}, find the opposite of each term.

$$6-6\sqrt{2}-11+6\sqrt{2}$$

Subtract 11 from 6 to get -5.

$$-5-6\sqrt{2}+6\sqrt{2}$$

Combine -6\sqrt{2} and 6\sqrt{2} to get 0.

$$-5$$

Factor

$$-5$$