Solve 7/-10-sqrt{2}

$$\frac { 7 } { - 10 - \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{2}-10}{14}\approx -0.61327046$$

Solution Steps

Rationalize the denominator of \frac{7}{-10-\sqrt{2}} by multiplying numerator and denominator by -10+\sqrt{2}.

$$\frac{7\left(-10+\sqrt{2}\right)}{\left(-10-\sqrt{2}\right)\left(-10+\sqrt{2}\right)}$$

Consider \left(-10-\sqrt{2}\right)\left(-10+\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}.

$$\frac{7\left(-10+\sqrt{2}\right)}{\left(-10\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square -10. Square \sqrt{2}.

$$\frac{7\left(-10+\sqrt{2}\right)}{100-2}$$

Subtract 2 from 100 to get 98.

$$\frac{7\left(-10+\sqrt{2}\right)}{98}$$

Divide 7\left(-10+\sqrt{2}\right) by 98 to get \frac{1}{14}\left(-10+\sqrt{2}\right).

$$\frac{1}{14}\left(-10+\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{14} by -10+\sqrt{2}.

$$\frac{1}{14}\left(-10\right)+\frac{1}{14}\sqrt{2}$$

Multiply \frac{1}{14} and -10 to get \frac{-10}{14}.

$$\frac{-10}{14}+\frac{1}{14}\sqrt{2}$$

Reduce the fraction \frac{-10}{14} to lowest terms by extracting and canceling out 2.

$$-\frac{5}{7}+\frac{1}{14}\sqrt{2}$$