Solve sqrt{18}*sqrt{12}-sqrt{8}-sqrt{27}+3sqrt{1/3}

$$\sqrt { 18 } \times \sqrt { 12 } - \sqrt { 8 } - \sqrt { 27 } + 3 \sqrt { \frac { 1 } { 3 } }$$

Evaluate

$$6\sqrt{6}-2\sqrt{2}-2\sqrt{3}\approx 8.404409717$$

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}.

$$3\sqrt{2}\sqrt{12}-\sqrt{8}-\sqrt{27}+3\sqrt{\frac{1}{3}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$3\sqrt{2}\times 2\sqrt{3}-\sqrt{8}-\sqrt{27}+3\sqrt{\frac{1}{3}}$$

Multiply 3 and 2 to get 6.

$$6\sqrt{2}\sqrt{3}-\sqrt{8}-\sqrt{27}+3\sqrt{\frac{1}{3}}$$

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

$$6\sqrt{6}-\sqrt{8}-\sqrt{27}+3\sqrt{\frac{1}{3}}$$

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}.

$$6\sqrt{6}-2\sqrt{2}-\sqrt{27}+3\sqrt{\frac{1}{3}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

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

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

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

Calculate the square root of 1 and get 1.

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

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

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

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