Solve frac{sqrt{5}+1}{sqrt{5}+2}+frac{sqrt{5}-1}{sqrt{5}-2})*2/sqrt{125}

$$\frac { \sqrt { 5 } + 1 } { \sqrt { 5 } + 2 } + \frac { \sqrt { 5 } - 1 } { \sqrt { 5 } - 2 } ) \times \frac { 2 } { \sqrt { 125 } }$$

Evaluate

$$6$$

Solution Steps

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

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

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}.

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

Square \sqrt{5}. Square 2.

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

Subtract 4 from 5 to get 1.

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

Anything divided by one gives itself.

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

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

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

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}.

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

Square \sqrt{5}. Square 2.

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

Subtract 4 from 5 to get 1.

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

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of \sqrt{5}+1 by each term of \sqrt{5}-2.

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

The square of \sqrt{5} is 5.

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

Combine -2\sqrt{5} and \sqrt{5} to get -\sqrt{5}.

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

Subtract 2 from 5 to get 3.

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

Apply the distributive property by multiplying each term of \sqrt{5}-1 by each term of \sqrt{5}+2.

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

The square of \sqrt{5} is 5.

$$3-\sqrt{5}+5+2\sqrt{5}-\sqrt{5}-2$$

Combine 2\sqrt{5} and -\sqrt{5} to get \sqrt{5}.

$$3-\sqrt{5}+5+\sqrt{5}-2$$

Subtract 2 from 5 to get 3.

$$3-\sqrt{5}+3+\sqrt{5}$$

Add 3 and 3 to get 6.

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

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

$$6$$

Factor

$$2\times 3$$