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

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

Evaluate

$$\frac{2\sqrt{3}\left(\sqrt{6}-1\right)}{5}\approx 1.004235952$$

Solution Steps

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

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

Consider \left(4\sqrt{3}+2\sqrt{2}\right)\left(4\sqrt{3}-2\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}.

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

Expand \left(4\sqrt{3}\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)}{16\left(\sqrt{3}\right)^{2}-\left(2\sqrt{2}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)}{16\times 3-\left(2\sqrt{2}\right)^{2}}$$

Multiply 16 and 3 to get 48.

$$\frac{4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)}{48-\left(2\sqrt{2}\right)^{2}}$$

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

$$\frac{4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)}{48-2^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)}{48-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)}{48-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)}{48-8}$$

Subtract 8 from 48 to get 40.

$$\frac{4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)}{40}$$

Divide 4\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right) by 40 to get \frac{1}{10}\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right).

$$\frac{1}{10}\sqrt{6}\left(4\sqrt{3}-2\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{10}\sqrt{6} by 4\sqrt{3}-2\sqrt{2}.

$$\frac{1}{10}\sqrt{6}\times 4\sqrt{3}+\frac{1}{10}\sqrt{6}\left(-2\right)\sqrt{2}$$

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

$$\frac{1}{10}\sqrt{3}\sqrt{2}\times 4\sqrt{3}+\frac{1}{10}\sqrt{6}\left(-2\right)\sqrt{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{1}{10}\times 3\times 4\sqrt{2}+\frac{1}{10}\sqrt{6}\left(-2\right)\sqrt{2}$$

Multiply \frac{1}{10} and 3 to get \frac{3}{10}.

$$\frac{3}{10}\times 4\sqrt{2}+\frac{1}{10}\sqrt{6}\left(-2\right)\sqrt{2}$$

Express \frac{3}{10}\times 4 as a single fraction.

$$\frac{3\times 4}{10}\sqrt{2}+\frac{1}{10}\sqrt{6}\left(-2\right)\sqrt{2}$$

Multiply 3 and 4 to get 12.

$$\frac{12}{10}\sqrt{2}+\frac{1}{10}\sqrt{6}\left(-2\right)\sqrt{2}$$

Reduce the fraction \frac{12}{10} to lowest terms by extracting and canceling out 2.

$$\frac{6}{5}\sqrt{2}+\frac{1}{10}\sqrt{6}\left(-2\right)\sqrt{2}$$

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}.

$$\frac{6}{5}\sqrt{2}+\frac{1}{10}\sqrt{2}\sqrt{3}\left(-2\right)\sqrt{2}$$

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

$$\frac{6}{5}\sqrt{2}+\frac{1}{10}\times 2\left(-2\right)\sqrt{3}$$

Multiply \frac{1}{10} and 2 to get \frac{2}{10}.

$$\frac{6}{5}\sqrt{2}+\frac{2}{10}\left(-2\right)\sqrt{3}$$

Reduce the fraction \frac{2}{10} to lowest terms by extracting and canceling out 2.

$$\frac{6}{5}\sqrt{2}+\frac{1}{5}\left(-2\right)\sqrt{3}$$

Multiply \frac{1}{5} and -2 to get \frac{-2}{5}.

$$\frac{6}{5}\sqrt{2}+\frac{-2}{5}\sqrt{3}$$

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

$$\frac{6}{5}\sqrt{2}-\frac{2}{5}\sqrt{3}$$