Solve 2/sqrt{6-2}-1/sqrt{5-2}

$$\frac{ 2 }{ \sqrt{ 6 } -2 } - \frac{ 1 }{ \sqrt{ 5 } -2 }$$

Evaluate

$$\sqrt{6}-\sqrt{5}\approx 0.213421765$$

Solution Steps

Rationalize the denominator of \frac{2}{\sqrt{6}-2} by multiplying numerator and denominator by \sqrt{6}+2.

$$\frac{2\left(\sqrt{6}+2\right)}{\left(\sqrt{6}-2\right)\left(\sqrt{6}+2\right)}-\frac{1}{\sqrt{5}-2}$$

Consider \left(\sqrt{6}-2\right)\left(\sqrt{6}+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(\sqrt{6}+2\right)}{\left(\sqrt{6}\right)^{2}-2^{2}}-\frac{1}{\sqrt{5}-2}$$

Square \sqrt{6}. Square 2.

$$\frac{2\left(\sqrt{6}+2\right)}{6-4}-\frac{1}{\sqrt{5}-2}$$

Subtract 4 from 6 to get 2.

$$\frac{2\left(\sqrt{6}+2\right)}{2}-\frac{1}{\sqrt{5}-2}$$

Cancel out 2 and 2.

$$\sqrt{6}+2-\frac{1}{\sqrt{5}-2}$$

Rationalize the denominator of \frac{1}{\sqrt{5}-2} by multiplying numerator and denominator by \sqrt{5}+2.

$$\sqrt{6}+2-\frac{\sqrt{5}+2}{\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)}$$

Consider \left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\sqrt{6}+2-\frac{\sqrt{5}+2}{\left(\sqrt{5}\right)^{2}-2^{2}}$$

Square \sqrt{5}. Square 2.

$$\sqrt{6}+2-\frac{\sqrt{5}+2}{5-4}$$

Subtract 4 from 5 to get 1.

$$\sqrt{6}+2-\frac{\sqrt{5}+2}{1}$$

Anything divided by one gives itself.

$$\sqrt{6}+2-\left(\sqrt{5}+2\right)$$

To find the opposite of \sqrt{5}+2, find the opposite of each term.

$$\sqrt{6}+2-\sqrt{5}-2$$

Subtract 2 from 2 to get 0.

$$\sqrt{6}-\sqrt{5}$$