Solve 5/3+sqrt{2}

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

Evaluate

$$\frac{15-5\sqrt{2}}{7}\approx 1.132704598$$

Solution Steps

Rationalize the denominator of \frac{5}{3+\sqrt{2}} by multiplying numerator and denominator by 3-\sqrt{2}.

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

Consider \left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 3. Square \sqrt{2}.

$$\frac{5\left(3-\sqrt{2}\right)}{9-2}$$

Subtract 2 from 9 to get 7.

$$\frac{5\left(3-\sqrt{2}\right)}{7}$$

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

$$\frac{15-5\sqrt{2}}{7}$$