Solve 7/-5+sqrt{2}

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

Evaluate

$$\frac{-7\sqrt{2}-35}{23}\approx -1.952151954$$

Solution Steps

Rationalize the denominator of \frac{7}{-5+\sqrt{2}} by multiplying numerator and denominator by -5-\sqrt{2}.

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

Consider \left(-5+\sqrt{2}\right)\left(-5-\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}.

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

Square -5. Square \sqrt{2}.

$$\frac{7\left(-5-\sqrt{2}\right)}{25-2}$$

Subtract 2 from 25 to get 23.

$$\frac{7\left(-5-\sqrt{2}\right)}{23}$$

Use the distributive property to multiply 7 by -5-\sqrt{2}.

$$\frac{-35-7\sqrt{2}}{23}$$