Solve left(2sqrt{5}-2sqrt{3}right)left(sqrt{12}+sqrt{20}right)

$$\left( 2 \sqrt{ 5 } -2 \sqrt{ 3 } \right) \left( \sqrt{ 12 } + \sqrt{ 20 } \right)$$

Evaluate

$$8$$

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}.

$$\left(2\sqrt{5}-2\sqrt{3}\right)\left(2\sqrt{3}+\sqrt{20}\right)$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\left(2\sqrt{5}-2\sqrt{3}\right)\left(2\sqrt{3}+2\sqrt{5}\right)$$

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

$$\left(2\sqrt{5}\right)^{2}-\left(2\sqrt{3}\right)^{2}$$

Expand \left(2\sqrt{5}\right)^{2}.

$$2^{2}\left(\sqrt{5}\right)^{2}-\left(2\sqrt{3}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{5}\right)^{2}-\left(2\sqrt{3}\right)^{2}$$

The square of \sqrt{5} is 5.

$$4\times 5-\left(2\sqrt{3}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$20-\left(2\sqrt{3}\right)^{2}$$

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

$$20-2^{2}\left(\sqrt{3}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$20-4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$20-4\times 3$$

Multiply 4 and 3 to get 12.

$$20-12$$

Subtract 12 from 20 to get 8.

$$8$$

Factor

$$2^{3}$$