Solve 5sqrt{12}+1/2sqrt{48}-6sqrt{2/3}*sqrt{2}

$$5 \sqrt { 12 } + \frac { 1 } { 2 } \sqrt { 48 } - 6 \sqrt { \frac { 2 } { 3 } } \times \sqrt { 2 }$$

Evaluate

$$8\sqrt{3}\approx 13.856406461$$

Solution Steps

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}.

$$5\times 2\sqrt{3}+\frac{1}{2}\sqrt{48}-6\sqrt{\frac{2}{3}}\sqrt{2}$$

Multiply 5 and 2 to get 10.

$$10\sqrt{3}+\frac{1}{2}\sqrt{48}-6\sqrt{\frac{2}{3}}\sqrt{2}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$10\sqrt{3}+\frac{1}{2}\times 4\sqrt{3}-6\sqrt{\frac{2}{3}}\sqrt{2}$$

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

$$10\sqrt{3}+\frac{4}{2}\sqrt{3}-6\sqrt{\frac{2}{3}}\sqrt{2}$$

Divide 4 by 2 to get 2.

$$10\sqrt{3}+2\sqrt{3}-6\sqrt{\frac{2}{3}}\sqrt{2}$$

Combine 10\sqrt{3} and 2\sqrt{3} to get 12\sqrt{3}.

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

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

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

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

$$12\sqrt{3}-6\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{2}$$

The square of \sqrt{3} is 3.

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

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

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

Cancel out 3, the greatest common factor in 6 and 3.

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

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$12\sqrt{3}-2\sqrt{2}\sqrt{3}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$12\sqrt{3}-2\times 2\sqrt{3}$$

Multiply 2 and 2 to get 4.

$$12\sqrt{3}-4\sqrt{3}$$

Combine 12\sqrt{3} and -4\sqrt{3} to get 8\sqrt{3}.

$$8\sqrt{3}$$