Solve (sqrt{5}+sqrt{3})^2-2sqrt{30}*sqrt{1/2}

$$( \sqrt { 5 } + \sqrt { 3 } ) ^ { 2 } - 2 \sqrt { 30 } \cdot \sqrt { \frac { 1 } { 2 } }$$

Evaluate

$$8$$

Solution Steps

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

$$\left(\sqrt{5}\right)^{2}+2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}-2\sqrt{30}\sqrt{\frac{1}{2}}$$

The square of \sqrt{5} is 5.

$$5+2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}-2\sqrt{30}\sqrt{\frac{1}{2}}$$

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

$$5+2\sqrt{15}+\left(\sqrt{3}\right)^{2}-2\sqrt{30}\sqrt{\frac{1}{2}}$$

The square of \sqrt{3} is 3.

$$5+2\sqrt{15}+3-2\sqrt{30}\sqrt{\frac{1}{2}}$$

Add 5 and 3 to get 8.

$$8+2\sqrt{15}-2\sqrt{30}\sqrt{\frac{1}{2}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$8+2\sqrt{15}-2\sqrt{30}\times \frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$8+2\sqrt{15}-2\sqrt{30}\times \frac{1}{\sqrt{2}}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$8+2\sqrt{15}-2\sqrt{30}\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$8+2\sqrt{15}-2\sqrt{30}\times \frac{\sqrt{2}}{2}$$

Cancel out 2 and 2.

$$8+2\sqrt{15}-\sqrt{2}\sqrt{30}$$

Factor 30=2\times 15. Rewrite the square root of the product \sqrt{2\times 15} as the product of square roots \sqrt{2}\sqrt{15}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Combine 2\sqrt{15} and -2\sqrt{15} to get 0.

$$8$$

Factor

$$2^{3}$$