Solve sqrt{18}-sqrt{1/2}divsqrt{11/3}*frac{6/sqrt{3}}{6/6}

$$\sqrt { 18 } - \sqrt { \frac { 1 } { 2 } } \div \sqrt { 1 \frac { 1 } { 3 } } \times \frac { \frac { 6 } { \sqrt { 3 } } } { \frac { 6 } { 6 } }$$

Evaluate

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

Solution Steps

Divide 6 by 6 to get 1.

$$\sqrt{18}-\frac{\sqrt{\frac{1}{2}}}{\sqrt{\frac{1\times 3+1}{3}}}\times \frac{\frac{6}{\sqrt{3}}}{1}$$

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}.

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

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

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

Calculate the square root of 1 and get 1.

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

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Multiply 1 and 3 to get 3.

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

Add 3 and 1 to get 4.

$$3\sqrt{2}-\frac{\frac{\sqrt{2}}{2}}{\sqrt{\frac{4}{3}}}\times \frac{\frac{6}{\sqrt{3}}}{1}$$

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$3\sqrt{2}-\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{4}}{\sqrt{3}}}\times \frac{\frac{6}{\sqrt{3}}}{1}$$

Calculate the square root of 4 and get 2.

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

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

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

The square of \sqrt{3} is 3.

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

Divide \frac{\sqrt{2}}{2} by \frac{2\sqrt{3}}{3} by multiplying \frac{\sqrt{2}}{2} by the reciprocal of \frac{2\sqrt{3}}{3}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Multiply 2 and 2 to get 4.

$$3\sqrt{2}-\frac{\sqrt{6}\times 3}{4\times 3}\times \frac{\frac{6}{\sqrt{3}}}{1}$$

Multiply 4 and 3 to get 12.

$$3\sqrt{2}-\frac{\sqrt{6}\times 3}{12}\times \frac{\frac{6}{\sqrt{3}}}{1}$$

Divide \sqrt{6}\times 3 by 12 to get \sqrt{6}\times \frac{1}{4}.

$$3\sqrt{2}-\sqrt{6}\times \frac{1}{4}\times \frac{\frac{6}{\sqrt{3}}}{1}$$

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

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

The square of \sqrt{3} is 3.

$$3\sqrt{2}-\sqrt{6}\times \frac{1}{4}\times \frac{\frac{6\sqrt{3}}{3}}{1}$$

Divide 6\sqrt{3} by 3 to get 2\sqrt{3}.

$$3\sqrt{2}-\sqrt{6}\times \frac{1}{4}\times \frac{2\sqrt{3}}{1}$$

Anything divided by one gives itself.

$$3\sqrt{2}-\sqrt{6}\times \frac{1}{4}\times 2\sqrt{3}$$

Multiply \frac{1}{4} and 2 to get \frac{2}{4}.

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

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

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

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}.

$$3\sqrt{2}-\sqrt{3}\sqrt{2}\times \frac{1}{2}\sqrt{3}$$

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

$$3\sqrt{2}-3\times \frac{1}{2}\sqrt{2}$$

Multiply 3 and \frac{1}{2} to get \frac{3}{2}.

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

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

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