Solve sqrt{18}-sqrt{9/2}-frac{sqrt{3}+sqrt{6}}{sqrt{3}}+(sqrt{3}-sqrt{2})^0+sqrt{(-sqrt{2})^2}

$$\sqrt { 18 } - \sqrt { \frac { 9 } { 2 } } - \frac { \sqrt { 3 } + \sqrt { 6 } } { \sqrt { 3 } } + ( \sqrt { 3 } - \sqrt { 2 } ) ^ { 0 } + \sqrt { ( - \sqrt { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{3\sqrt{2}}{2}\approx 2.121320344$$

Short 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}.

$$3\sqrt{2}-\sqrt{\frac{9}{2}}-\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\left(\sqrt{3}-\sqrt{2}\right)^{0}+\sqrt{\left(-\sqrt{2}\right)^{2}}$$

Rewrite the square root of the division \sqrt{\frac{9}{2}} as the division of square roots \frac{\sqrt{9}}{\sqrt{2}}.

$$3\sqrt{2}-\frac{\sqrt{9}}{\sqrt{2}}-\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\left(\sqrt{3}-\sqrt{2}\right)^{0}+\sqrt{\left(-\sqrt{2}\right)^{2}}$$

Calculate the square root of 9 and get 3.

$$3\sqrt{2}-\frac{3}{\sqrt{2}}-\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\left(\sqrt{3}-\sqrt{2}\right)^{0}+\sqrt{\left(-\sqrt{2}\right)^{2}}$$

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

$$3\sqrt{2}-\frac{3\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\left(\sqrt{3}-\sqrt{2}\right)^{0}+\sqrt{\left(-\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$3\sqrt{2}-\frac{3\sqrt{2}}{2}-\frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}}+\left(\sqrt{3}-\sqrt{2}\right)^{0}+\sqrt{\left(-\sqrt{2}\right)^{2}}$$

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

$$3\sqrt{2}-\frac{3\sqrt{2}}{2}-\frac{\left(\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\left(\sqrt{3}-\sqrt{2}\right)^{0}+\sqrt{\left(-\sqrt{2}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$3\sqrt{2}-\frac{3\sqrt{2}}{2}-\frac{\left(\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{3}+\left(\sqrt{3}-\sqrt{2}\right)^{0}+\sqrt{\left(-\sqrt{2}\right)^{2}}$$

Calculate \sqrt{3}-\sqrt{2} to the power of 0 and get 1.

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

Calculate -\sqrt{2} to the power of 2 and get \left(\sqrt{2}\right)^{2}.

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{3\sqrt{2}+2}{2} times \frac{3}{3}. Multiply \frac{\left(\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{3} times \frac{2}{2}.

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

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

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

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

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

Do the calculations in 9\sqrt{2}+6-6-6\sqrt{2}.

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

Use the distributive property to multiply \sqrt{3}+\sqrt{6} by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

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}.

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

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

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

Divide each term of 3+3\sqrt{2} by 3 to get 1+\sqrt{2}.

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

To find the opposite of 1+\sqrt{2}, find the opposite of each term.

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

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

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

Add -1 and 1 to get 0.

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

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

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

Since \frac{2\times 2\sqrt{2}}{2} and \frac{3\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\times 2\sqrt{2}-3\sqrt{2}.

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

Do the calculations in 4\sqrt{2}-3\sqrt{2}.

$$\frac{\sqrt{2}}{2}+\sqrt{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

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

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

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

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

$$\frac{2\times 3\sqrt{2}}{2}-\frac{3\sqrt{2}}{2}$$

Since \frac{2\times 3\sqrt{2}}{2} and \frac{3\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\times 3\sqrt{2}-3\sqrt{2}}{2}$$

Do the multiplications in 2\times 3\sqrt{2}-3\sqrt{2}.

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

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

$$\frac{3\sqrt{2}}{2}$$