Solve 7/2+sqrt{13}

$$\frac{ 7 }{ 2+ \sqrt{ 13 } }$$

Evaluate

$$\frac{7\left(\sqrt{13}-2\right)}{9}\approx 1.248762103$$

Solution Steps

Rationalize the denominator of \frac{7}{2+\sqrt{13}} by multiplying numerator and denominator by 2-\sqrt{13}.

$$\frac{7\left(2-\sqrt{13}\right)}{\left(2+\sqrt{13}\right)\left(2-\sqrt{13}\right)}$$

Consider \left(2+\sqrt{13}\right)\left(2-\sqrt{13}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{7\left(2-\sqrt{13}\right)}{2^{2}-\left(\sqrt{13}\right)^{2}}$$

Square 2. Square \sqrt{13}.

$$\frac{7\left(2-\sqrt{13}\right)}{4-13}$$

Subtract 13 from 4 to get -9.

$$\frac{7\left(2-\sqrt{13}\right)}{-9}$$

Use the distributive property to multiply 7 by 2-\sqrt{13}.

$$\frac{14-7\sqrt{13}}{-9}$$