Solve sqrt{12}(sqrt{75}+3sqrt{1/3}-sqrt{48})

$$\sqrt { 12 } ( \sqrt { 75 } + 3 \sqrt { \frac { 1 } { 3 } } - \sqrt { 48 } )$$

Evaluate

$$12$$

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}.

$$2\sqrt{3}\left(\sqrt{75}+3\sqrt{\frac{1}{3}}-\sqrt{48}\right)$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$2\sqrt{3}\left(5\sqrt{3}+3\sqrt{\frac{1}{3}}-\sqrt{48}\right)$$

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

$$2\sqrt{3}\left(5\sqrt{3}+3\times \frac{\sqrt{1}}{\sqrt{3}}-\sqrt{48}\right)$$

Calculate the square root of 1 and get 1.

$$2\sqrt{3}\left(5\sqrt{3}+3\times \frac{1}{\sqrt{3}}-\sqrt{48}\right)$$

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

$$2\sqrt{3}\left(5\sqrt{3}+3\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\sqrt{48}\right)$$

The square of \sqrt{3} is 3.

$$2\sqrt{3}\left(5\sqrt{3}+3\times \frac{\sqrt{3}}{3}-\sqrt{48}\right)$$

Cancel out 3 and 3.

$$2\sqrt{3}\left(5\sqrt{3}+\sqrt{3}-\sqrt{48}\right)$$

Combine 5\sqrt{3} and \sqrt{3} to get 6\sqrt{3}.

$$2\sqrt{3}\left(6\sqrt{3}-\sqrt{48}\right)$$

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}.

$$2\sqrt{3}\left(6\sqrt{3}-4\sqrt{3}\right)$$

Combine 6\sqrt{3} and -4\sqrt{3} to get 2\sqrt{3}.

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

Multiply 2 and 2 to get 4.

$$4\sqrt{3}\sqrt{3}$$

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

$$4\times 3$$

Multiply 4 and 3 to get 12.

$$12$$

Factor

$$2^{2}\times 3$$