Solve frac{18-3sqrt{2}m}{3sqrt{2}-m}

$$\frac { 18 - 3 \sqrt { 2 } m } { 3 \sqrt { 2 } - m }$$

Evaluate

$$\frac{3\left(-\sqrt{2}m+6\right)}{-m+3\sqrt{2}}$$

Short Solution Steps

Rationalize the denominator of \frac{18-3\sqrt{2}m}{3\sqrt{2}-m} by multiplying numerator and denominator by 3\sqrt{2}+m.

$$\frac{\left(18-3\sqrt{2}m\right)\left(3\sqrt{2}+m\right)}{\left(3\sqrt{2}-m\right)\left(3\sqrt{2}+m\right)}$$

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

$$\frac{\left(18-3\sqrt{2}m\right)\left(3\sqrt{2}+m\right)}{\left(3\sqrt{2}\right)^{2}-m^{2}}$$

Expand \left(3\sqrt{2}\right)^{2}.

$$\frac{\left(18-3\sqrt{2}m\right)\left(3\sqrt{2}+m\right)}{3^{2}\left(\sqrt{2}\right)^{2}-m^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(18-3\sqrt{2}m\right)\left(3\sqrt{2}+m\right)}{9\left(\sqrt{2}\right)^{2}-m^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(18-3\sqrt{2}m\right)\left(3\sqrt{2}+m\right)}{9\times 2-m^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{\left(18-3\sqrt{2}m\right)\left(3\sqrt{2}+m\right)}{18-m^{2}}$$

Use the distributive property to multiply 18-3\sqrt{2}m by 3\sqrt{2}+m.

$$\frac{3\left(18-3\sqrt{2}m\right)\sqrt{2}+\left(18-3\sqrt{2}m\right)m}{18-m^{2}}$$

Use the distributive property to multiply 3 by 18-3\sqrt{2}m.

$$\frac{\left(54-9m\sqrt{2}\right)\sqrt{2}+\left(18-3\sqrt{2}m\right)m}{18-m^{2}}$$

Use the distributive property to multiply 54-9m\sqrt{2} by \sqrt{2}.

$$\frac{54\sqrt{2}-9m\left(\sqrt{2}\right)^{2}+\left(18-3\sqrt{2}m\right)m}{18-m^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{54\sqrt{2}-9m\times 2+\left(18-3\sqrt{2}m\right)m}{18-m^{2}}$$

Multiply -9 and 2 to get -18.

$$\frac{54\sqrt{2}-18m+\left(18-3\sqrt{2}m\right)m}{18-m^{2}}$$

Use the distributive property to multiply 18-3\sqrt{2}m by m.

$$\frac{54\sqrt{2}-18m+18m-3\sqrt{2}m^{2}}{18-m^{2}}$$

Combine -18m and 18m to get 0.

$$\frac{54\sqrt{2}-3\sqrt{2}m^{2}}{18-m^{2}}$$

Factor the expressions that are not already factored.

$$\frac{3\sqrt{2}\left(-m^{2}+18\right)}{-m^{2}+18}$$

Cancel out -m^{2}+18 in both numerator and denominator.

$$3\sqrt{2}$$

Factor

$$\frac{3\left(-\sqrt{2}m+6\right)}{-m+3\sqrt{2}}$$