Solve frac{7-sqrt{9}}{3-sqrt{2}}

$$\frac { 7 - \sqrt { 9 } } { 3 - \sqrt { 2 } }$$

Evaluate

$$\frac{4\left(\sqrt{2}+3\right)}{7}\approx 2.52240775$$

Solution Steps

Calculate the square root of 9 and get 3.

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

Subtract 3 from 7 to get 4.

$$\frac{4}{3-\sqrt{2}}$$

Rationalize the denominator of \frac{4}{3-\sqrt{2}} by multiplying numerator and denominator by 3+\sqrt{2}.

$$\frac{4\left(3+\sqrt{2}\right)}{\left(3-\sqrt{2}\right)\left(3+\sqrt{2}\right)}$$

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}.

$$\frac{4\left(3+\sqrt{2}\right)}{3^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 3. Square \sqrt{2}.

$$\frac{4\left(3+\sqrt{2}\right)}{9-2}$$

Subtract 2 from 9 to get 7.

$$\frac{4\left(3+\sqrt{2}\right)}{7}$$

Use the distributive property to multiply 4 by 3+\sqrt{2}.

$$\frac{12+4\sqrt{2}}{7}$$