Solve 03+5/1+sqrt{5}

$$\frac { 03 + 5 } { 1 + \sqrt { 5 } }$$

Evaluate

$$\frac{5\left(\sqrt{5}-1\right)}{4}\approx 1.545084972$$

Solution Steps

Multiply 0 and 3 to get 0.

$$\frac{0+5}{1+\sqrt{5}}$$

Add 0 and 5 to get 5.

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

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

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

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

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

Square 1. Square \sqrt{5}.

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

Subtract 5 from 1 to get -4.

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

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

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

Factor

$$\frac{5 {(\sqrt{5} - 1)}}{4} = 1.5450849718747373$$