Solve 4/sqrt{5-1}-4/sqrt{5+1}?

$$\frac { 4 } { \sqrt { 5 } - 1 } - \frac { 4 } { \sqrt { 5 } + 1 } ?$$

Evaluate

$$2$$

Solution Steps

Rationalize the denominator of \frac{4}{\sqrt{5}-1} by multiplying numerator and denominator by \sqrt{5}+1.

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

Consider \left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square 1.

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

Subtract 1 from 5 to get 4.

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

Cancel out 4 and 4.

$$\sqrt{5}+1-\frac{4}{\sqrt{5}+1}$$

Rationalize the denominator of \frac{4}{\sqrt{5}+1} by multiplying numerator and denominator by \sqrt{5}-1.

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

Consider \left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square 1.

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

Subtract 1 from 5 to get 4.

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

Cancel out 4 and 4.

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

To find the opposite of \sqrt{5}-1, find the opposite of each term.

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

The opposite of -1 is 1.

$$\sqrt{5}+1-\sqrt{5}+1$$

Combine \sqrt{5} and -\sqrt{5} to get 0.

$$1+1$$

Add 1 and 1 to get 2.

$$2$$

Factor

$$2$$