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

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

Evaluate

$$\frac{12-6\sqrt{3}}{5}\approx 0.321539031$$

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}.

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

Multiply 4 and 3 to get 12.

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

Combine 3\sqrt{2} and 2\sqrt{2} to get 5\sqrt{2}.

$$\frac{12\sqrt{2}-6\sqrt{6}}{5\sqrt{2}}$$

Rationalize the denominator of \frac{12\sqrt{2}-6\sqrt{6}}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\sqrt{2}}{5\times 2}$$

Multiply 5 and 2 to get 10.

$$\frac{\left(12\sqrt{2}-6\sqrt{6}\right)\sqrt{2}}{10}$$

Use the distributive property to multiply 12\sqrt{2}-6\sqrt{6} by \sqrt{2}.

$$\frac{12\left(\sqrt{2}\right)^{2}-6\sqrt{6}\sqrt{2}}{10}$$

The square of \sqrt{2} is 2.

$$\frac{12\times 2-6\sqrt{6}\sqrt{2}}{10}$$

Multiply 12 and 2 to get 24.

$$\frac{24-6\sqrt{6}\sqrt{2}}{10}$$

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}.

$$\frac{24-6\sqrt{2}\sqrt{3}\sqrt{2}}{10}$$

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

$$\frac{24-6\times 2\sqrt{3}}{10}$$

Multiply -6 and 2 to get -12.

$$\frac{24-12\sqrt{3}}{10}$$

Factor

$$\frac{6 {(2 - \sqrt{3})}}{5} = 0.32153903091734737$$