Solve 48/5+sqrt{13}

$$\frac { 48 } { 5 + \sqrt { 13 } }$$

Evaluate

$$20-4\sqrt{13}\approx 5.577794898$$

Solution Steps

Rationalize the denominator of \frac{48}{5+\sqrt{13}} by multiplying numerator and denominator by 5-\sqrt{13}.

$$\frac{48\left(5-\sqrt{13}\right)}{\left(5+\sqrt{13}\right)\left(5-\sqrt{13}\right)}$$

Consider \left(5+\sqrt{13}\right)\left(5-\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}.

$$\frac{48\left(5-\sqrt{13}\right)}{5^{2}-\left(\sqrt{13}\right)^{2}}$$

Square 5. Square \sqrt{13}.

$$\frac{48\left(5-\sqrt{13}\right)}{25-13}$$

Subtract 13 from 25 to get 12.

$$\frac{48\left(5-\sqrt{13}\right)}{12}$$

Divide 48\left(5-\sqrt{13}\right) by 12 to get 4\left(5-\sqrt{13}\right).

$$4\left(5-\sqrt{13}\right)$$

Use the distributive property to multiply 4 by 5-\sqrt{13}.

$$20-4\sqrt{13}$$