Solve 7/sqrt{2+3}

$$\frac { 7 } { \sqrt { 2 } + 3 }$$

Evaluate

$$3-\sqrt{2}\approx 1.585786438$$

Solution Steps

Rationalize the denominator of \frac{7}{\sqrt{2}+3} by multiplying numerator and denominator by \sqrt{2}-3.

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

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

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

Square \sqrt{2}. Square 3.

$$\frac{7\left(\sqrt{2}-3\right)}{2-9}$$

Subtract 9 from 2 to get -7.

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

Cancel out -7 and -7.

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

To find the opposite of \sqrt{2}-3, find the opposite of each term.

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

The opposite of -3 is 3.

$$-\sqrt{2}+3$$