Solve 1/2-sqrt{3}+1/2+sqrt{3}+frac{sqrt{8}}{sqrt{2}}

$$\frac { 1 } { 2 - \sqrt { 3 } } + \frac { 1 } { 2 + \sqrt { 3 } } + \frac { \sqrt { 8 } } { \sqrt { 2 } }$$

Evaluate

$$6$$

Solution Steps

Rationalize the denominator of \frac{1}{2-\sqrt{3}} by multiplying numerator and denominator by 2+\sqrt{3}.

$$\frac{2+\sqrt{3}}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}+\frac{1}{2+\sqrt{3}}+\frac{\sqrt{8}}{\sqrt{2}}$$

Consider \left(2-\sqrt{3}\right)\left(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}.

$$\frac{2+\sqrt{3}}{2^{2}-\left(\sqrt{3}\right)^{2}}+\frac{1}{2+\sqrt{3}}+\frac{\sqrt{8}}{\sqrt{2}}$$

Square 2. Square \sqrt{3}.

$$\frac{2+\sqrt{3}}{4-3}+\frac{1}{2+\sqrt{3}}+\frac{\sqrt{8}}{\sqrt{2}}$$

Subtract 3 from 4 to get 1.

$$\frac{2+\sqrt{3}}{1}+\frac{1}{2+\sqrt{3}}+\frac{\sqrt{8}}{\sqrt{2}}$$

Anything divided by one gives itself.

$$2+\sqrt{3}+\frac{1}{2+\sqrt{3}}+\frac{\sqrt{8}}{\sqrt{2}}$$

Rationalize the denominator of \frac{1}{2+\sqrt{3}} by multiplying numerator and denominator by 2-\sqrt{3}.

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

Consider \left(2+\sqrt{3}\right)\left(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}.

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

Square 2. Square \sqrt{3}.

$$2+\sqrt{3}+\frac{2-\sqrt{3}}{4-3}+\frac{\sqrt{8}}{\sqrt{2}}$$

Subtract 3 from 4 to get 1.

$$2+\sqrt{3}+\frac{2-\sqrt{3}}{1}+\frac{\sqrt{8}}{\sqrt{2}}$$

Anything divided by one gives itself.

$$2+\sqrt{3}+2-\sqrt{3}+\frac{\sqrt{8}}{\sqrt{2}}$$

Add 2 and 2 to get 4.

$$4+\sqrt{3}-\sqrt{3}+\frac{\sqrt{8}}{\sqrt{2}}$$

Combine \sqrt{3} and -\sqrt{3} to get 0.

$$4+\frac{\sqrt{8}}{\sqrt{2}}$$

Rewrite the division of square roots \frac{\sqrt{8}}{\sqrt{2}} as the square root of the division \sqrt{\frac{8}{2}} and perform the division.

$$4+\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$4+2$$

Add 4 and 2 to get 6.

$$6$$

Factor

$$2\times 3$$