Solve frac{4-3sqrt{6}}{3sqrt{6}-2}

$$\frac { 4 - 3 \sqrt { 6 } } { 3 \sqrt { 6 } - 2 }$$

Evaluate

$$\frac{3\sqrt{6}-23}{25}\approx -0.626061231$$

Solution Steps

Rationalize the denominator of \frac{4-3\sqrt{6}}{3\sqrt{6}-2} by multiplying numerator and denominator by 3\sqrt{6}+2.

$$\frac{\left(4-3\sqrt{6}\right)\left(3\sqrt{6}+2\right)}{\left(3\sqrt{6}-2\right)\left(3\sqrt{6}+2\right)}$$

Consider \left(3\sqrt{6}-2\right)\left(3\sqrt{6}+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}.

$$\frac{\left(4-3\sqrt{6}\right)\left(3\sqrt{6}+2\right)}{\left(3\sqrt{6}\right)^{2}-2^{2}}$$

Expand \left(3\sqrt{6}\right)^{2}.

$$\frac{\left(4-3\sqrt{6}\right)\left(3\sqrt{6}+2\right)}{3^{2}\left(\sqrt{6}\right)^{2}-2^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(4-3\sqrt{6}\right)\left(3\sqrt{6}+2\right)}{9\left(\sqrt{6}\right)^{2}-2^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(4-3\sqrt{6}\right)\left(3\sqrt{6}+2\right)}{9\times 6-2^{2}}$$

Multiply 9 and 6 to get 54.

$$\frac{\left(4-3\sqrt{6}\right)\left(3\sqrt{6}+2\right)}{54-2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(4-3\sqrt{6}\right)\left(3\sqrt{6}+2\right)}{54-4}$$

Subtract 4 from 54 to get 50.

$$\frac{\left(4-3\sqrt{6}\right)\left(3\sqrt{6}+2\right)}{50}$$

Apply the distributive property by multiplying each term of 4-3\sqrt{6} by each term of 3\sqrt{6}+2.

$$\frac{12\sqrt{6}+8-9\left(\sqrt{6}\right)^{2}-6\sqrt{6}}{50}$$

The square of \sqrt{6} is 6.

$$\frac{12\sqrt{6}+8-9\times 6-6\sqrt{6}}{50}$$

Multiply -9 and 6 to get -54.

$$\frac{12\sqrt{6}+8-54-6\sqrt{6}}{50}$$

Subtract 54 from 8 to get -46.

$$\frac{12\sqrt{6}-46-6\sqrt{6}}{50}$$

Combine 12\sqrt{6} and -6\sqrt{6} to get 6\sqrt{6}.

$$\frac{6\sqrt{6}-46}{50}$$