Solve frac{1+sqrt{25}}{sqrt{3}+sqrt{5}}=

$$\frac{ 1+ \sqrt{ 25 } }{ \sqrt{ 3 } + \sqrt{ 5 } } =$$

Evaluate

$$3\left(\sqrt{5}-\sqrt{3}\right)\approx 1.51205151$$

Solution Steps

Calculate the square root of 25 and get 5.

$$\frac{1+5}{\sqrt{3}+\sqrt{5}}$$

Add 1 and 5 to get 6.

$$\frac{6}{\sqrt{3}+\sqrt{5}}$$

Rationalize the denominator of \frac{6}{\sqrt{3}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{3}-\sqrt{5}.

$$\frac{6\left(\sqrt{3}-\sqrt{5}\right)}{\left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{5}\right)}$$

Consider \left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{3}-\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}.

$$\frac{6\left(\sqrt{3}-\sqrt{5}\right)}{\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Square \sqrt{3}. Square \sqrt{5}.

$$\frac{6\left(\sqrt{3}-\sqrt{5}\right)}{3-5}$$

Subtract 5 from 3 to get -2.

$$\frac{6\left(\sqrt{3}-\sqrt{5}\right)}{-2}$$

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

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

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

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