Solve 5/-3-sqrt{2}

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

Evaluate

$$\frac{5\left(\sqrt{2}-3\right)}{7}\approx -1.132704598$$

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}.

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

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}.

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