Solve left(3sqrt{2}+sqrt{48}right)left(sqrt{18}-4sqrt{3}right)

$$\left( 3 \sqrt{ 2 } + \sqrt{ 48 } \right) \left( \sqrt{ 18 } -4 \sqrt{ 3 } \right)$$

Evaluate

$$-30$$

Solution Steps

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

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

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}.

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{2}\right)^{2}-\left(4\sqrt{3}\right)^{2}$$

The square of \sqrt{2} is 2.

$$9\times 2-\left(4\sqrt{3}\right)^{2}$$

Multiply 9 and 2 to get 18.

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

Expand \left(4\sqrt{3}\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$18-16\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$18-16\times 3$$

Multiply 16 and 3 to get 48.

$$18-48$$

Subtract 48 from 18 to get -30.

$$-30$$

Factor

$$-30$$