Solve (3sqrt{2}-sqrt{12})quad(sqrt{18}+2sqrt{3})

$$( 3 \sqrt { 2 } - \sqrt { 12 } ) \quad ( \sqrt { 18 } + 2 \sqrt { 3 } )$$

Evaluate

$$6$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{2} is 2.

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

Multiply 9 and 2 to get 18.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

$$18-4\times 3$$

Multiply 4 and 3 to get 12.

$$18-12$$

Subtract 12 from 18 to get 6.

$$6$$

Factor

$$2\times 3$$