Solve (sqrt{3}+sqrt{5})^2-(sqrt{3}-sqrt{5})^2

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

Evaluate

$$4\sqrt{15}\approx 15.491933385$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3}+\sqrt{5}\right)^{2}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$3+2\sqrt{15}+\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$3+2\sqrt{15}+5-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

Add 3 and 5 to get 8.

$$8+2\sqrt{15}-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-\sqrt{5}\right)^{2}.

$$8+2\sqrt{15}-\left(\left(\sqrt{3}\right)^{2}-2\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$8+2\sqrt{15}-\left(3-2\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$8+2\sqrt{15}-\left(3-2\sqrt{15}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$8+2\sqrt{15}-\left(3-2\sqrt{15}+5\right)$$

Add 3 and 5 to get 8.

$$8+2\sqrt{15}-\left(8-2\sqrt{15}\right)$$

To find the opposite of 8-2\sqrt{15}, find the opposite of each term.

$$8+2\sqrt{15}-8+2\sqrt{15}$$

Subtract 8 from 8 to get 0.

$$2\sqrt{15}+2\sqrt{15}$$

Combine 2\sqrt{15} and 2\sqrt{15} to get 4\sqrt{15}.

$$4\sqrt{15}$$

Expand

$$4 \sqrt{15} = 15.491933385$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3}+\sqrt{5}\right)^{2}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$3+2\sqrt{15}+\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$3+2\sqrt{15}+5-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

Add 3 and 5 to get 8.

$$8+2\sqrt{15}-\left(\sqrt{3}-\sqrt{5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-\sqrt{5}\right)^{2}.

$$8+2\sqrt{15}-\left(\left(\sqrt{3}\right)^{2}-2\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$8+2\sqrt{15}-\left(3-2\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}\right)$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$8+2\sqrt{15}-\left(3-2\sqrt{15}+\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$8+2\sqrt{15}-\left(3-2\sqrt{15}+5\right)$$

Add 3 and 5 to get 8.

$$8+2\sqrt{15}-\left(8-2\sqrt{15}\right)$$

To find the opposite of 8-2\sqrt{15}, find the opposite of each term.

$$8+2\sqrt{15}-8+2\sqrt{15}$$

Subtract 8 from 8 to get 0.

$$2\sqrt{15}+2\sqrt{15}$$

Combine 2\sqrt{15} and 2\sqrt{15} to get 4\sqrt{15}.

$$4\sqrt{15}$$