Solve 6/sqrt{5+1}

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

Evaluate

$$\frac{3\left(\sqrt{5}-1\right)}{2}\approx 1.854101966$$

Solution Steps

Rationalize the denominator of \frac{6}{\sqrt{5}+1} by multiplying numerator and denominator by \sqrt{5}-1.

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

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}.

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

Square \sqrt{5}. Square 1.

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

Subtract 1 from 5 to get 4.

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

Divide 6\left(\sqrt{5}-1\right) by 4 to get \frac{3}{2}\left(\sqrt{5}-1\right).

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

Use the distributive property to multiply \frac{3}{2} by \sqrt{5}-1.

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

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

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