Solve 1/2-sqrt{3}*1/2+sqrt{3}

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

Evaluate

$$1$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

$$4-3$$

Subtract 3 from 4 to get 1.

$$1$$

Factor

$$1$$