Solve sqrt{3}-((3sqrt{3}-sqrt{8})-(sqrt{18}-sqrt{2}))

$$\sqrt{ 3 } -((3 \sqrt{ 3 } - \sqrt{ 8 } )-( \sqrt{ 18 } - \sqrt{ 2 } ))$$

Evaluate

$$2^{\frac{5}{2}}-2\sqrt{3}\approx 2.192752634$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\sqrt{3}-\left(3\sqrt{3}-2\sqrt{2}-\left(\sqrt{18}-\sqrt{2}\right)\right)$$

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}.

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

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

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

Combine -2\sqrt{2} and -2\sqrt{2} to get -4\sqrt{2}.

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

To find the opposite of 3\sqrt{3}-4\sqrt{2}, find the opposite of each term.

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

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

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

The opposite of -4\sqrt{2} is 4\sqrt{2}.

$$-2\sqrt{3}+4\sqrt{2}$$