Solve frac{4sqrt{18}-6sqrt{6}}{3sqrt{2}+2sqrt{6}}

$$\frac{ 4 \sqrt{ 18 } -6 \sqrt{ 6 } }{ 3 \sqrt{ 2 } +2 \sqrt{ 6 } }$$

Evaluate

$$14\sqrt{3}-24\approx 0.248711306$$

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

Multiply 4 and 3 to get 12.

$$\frac{12\sqrt{2}-6\sqrt{6}}{3\sqrt{2}+2\sqrt{6}}$$

Rationalize the denominator of \frac{12\sqrt{2}-6\sqrt{6}}{3\sqrt{2}+2\sqrt{6}} by multiplying numerator and denominator by 3\sqrt{2}-2\sqrt{6}.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{\left(3\sqrt{2}+2\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}$$

Consider \left(3\sqrt{2}+2\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{\left(3\sqrt{2}\right)^{2}-\left(2\sqrt{6}\right)^{2}}$$

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

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{3^{2}\left(\sqrt{2}\right)^{2}-\left(2\sqrt{6}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{9\left(\sqrt{2}\right)^{2}-\left(2\sqrt{6}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{9\times 2-\left(2\sqrt{6}\right)^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{18-\left(2\sqrt{6}\right)^{2}}$$

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

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{18-2^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{18-4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{18-4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{18-24}$$

Subtract 24 from 18 to get -6.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\left(3\sqrt{2}-2\sqrt{6}\right)}{-6}$$

Apply the distributive property by multiplying each term of 12\sqrt{2}-6\sqrt{6} by each term of 3\sqrt{2}-2\sqrt{6}.

$$\frac{36\left(\sqrt{2}\right)^{2}-24\sqrt{2}\sqrt{6}-18\sqrt{6}\sqrt{2}+12\left(\sqrt{6}\right)^{2}}{-6}$$

The square of \sqrt{2} is 2.

$$\frac{36\times 2-24\sqrt{2}\sqrt{6}-18\sqrt{6}\sqrt{2}+12\left(\sqrt{6}\right)^{2}}{-6}$$

Multiply 36 and 2 to get 72.

$$\frac{72-24\sqrt{2}\sqrt{6}-18\sqrt{6}\sqrt{2}+12\left(\sqrt{6}\right)^{2}}{-6}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\frac{72-24\sqrt{2}\sqrt{2}\sqrt{3}-18\sqrt{6}\sqrt{2}+12\left(\sqrt{6}\right)^{2}}{-6}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{72-24\times 2\sqrt{3}-18\sqrt{6}\sqrt{2}+12\left(\sqrt{6}\right)^{2}}{-6}$$

Multiply -24 and 2 to get -48.

$$\frac{72-48\sqrt{3}-18\sqrt{6}\sqrt{2}+12\left(\sqrt{6}\right)^{2}}{-6}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\frac{72-48\sqrt{3}-18\sqrt{2}\sqrt{3}\sqrt{2}+12\left(\sqrt{6}\right)^{2}}{-6}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{72-48\sqrt{3}-18\times 2\sqrt{3}+12\left(\sqrt{6}\right)^{2}}{-6}$$

Multiply -18 and 2 to get -36.

$$\frac{72-48\sqrt{3}-36\sqrt{3}+12\left(\sqrt{6}\right)^{2}}{-6}$$

Combine -48\sqrt{3} and -36\sqrt{3} to get -84\sqrt{3}.

$$\frac{72-84\sqrt{3}+12\left(\sqrt{6}\right)^{2}}{-6}$$

The square of \sqrt{6} is 6.

$$\frac{72-84\sqrt{3}+12\times 6}{-6}$$

Multiply 12 and 6 to get 72.

$$\frac{72-84\sqrt{3}+72}{-6}$$

Add 72 and 72 to get 144.

$$\frac{144-84\sqrt{3}}{-6}$$

Factor

$$2 {(7 \sqrt{3} - 12)} = 0.248711306$$