Solve 1+2sqrt{2}-left(2sqrt{2}-2right)frac{sqrt{3}-3}{3}

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

Evaluate

$$\frac{2\sqrt{3}}{3}-\frac{2\sqrt{6}}{3}+4\sqrt{2}-1\approx 4.178561626$$

Short Solution Steps

Express \left(2\sqrt{2}-2\right)\times \frac{\sqrt{3}-3}{3} as a single fraction.

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

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

$$1+2\sqrt{2}-\frac{2\sqrt{2}\sqrt{3}-6\sqrt{2}-2\sqrt{3}+6}{3}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$1+2\sqrt{2}-\frac{2\sqrt{6}-6\sqrt{2}-2\sqrt{3}+6}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1+2\sqrt{2} times \frac{3}{3}.

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

Since \frac{3\left(1+2\sqrt{2}\right)}{3} and \frac{2\sqrt{6}-6\sqrt{2}-2\sqrt{3}+6}{3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(1+2\sqrt{2}\right)-\left(2\sqrt{6}-6\sqrt{2}-2\sqrt{3}+6\right).

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

Do the calculations in 3+6\sqrt{2}-2\sqrt{6}+6\sqrt{2}+2\sqrt{3}-6.

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

Factor

$$\frac{2 \sqrt{3} + 12 \sqrt{2} - 2 \sqrt{6} - 3}{3} = 4.17856162601618$$