Solve 2sqrt{12}-18sqrt{1/27}+3sqrt{148}

$$2 \sqrt { 12 } - 18 \sqrt { \frac { 1 } { 27 } } + 3 \sqrt { 148 }$$

Evaluate

$$2\sqrt{3}\left(\sqrt{111}+1\right)\approx 39.960676797$$

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}.

$$2\times 2\sqrt{3}-18\sqrt{\frac{1}{27}}+3\sqrt{148}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{3}-18\sqrt{\frac{1}{27}}+3\sqrt{148}$$

Rewrite the square root of the division \sqrt{\frac{1}{27}} as the division of square roots \frac{\sqrt{1}}{\sqrt{27}}.

$$4\sqrt{3}-18\times \frac{\sqrt{1}}{\sqrt{27}}+3\sqrt{148}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{3}-18\times \frac{1}{\sqrt{27}}+3\sqrt{148}$$

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}.

$$4\sqrt{3}-18\times \frac{1}{3\sqrt{3}}+3\sqrt{148}$$

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

$$4\sqrt{3}-18\times \frac{\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}+3\sqrt{148}$$

The square of \sqrt{3} is 3.

$$4\sqrt{3}-18\times \frac{\sqrt{3}}{3\times 3}+3\sqrt{148}$$

Multiply 3 and 3 to get 9.

$$4\sqrt{3}-18\times \frac{\sqrt{3}}{9}+3\sqrt{148}$$

Cancel out 9, the greatest common factor in 18 and 9.

$$4\sqrt{3}-2\sqrt{3}+3\sqrt{148}$$

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

$$2\sqrt{3}+3\sqrt{148}$$

Factor 148=2^{2}\times 37. Rewrite the square root of the product \sqrt{2^{2}\times 37} as the product of square roots \sqrt{2^{2}}\sqrt{37}. Take the square root of 2^{2}.

$$2\sqrt{3}+3\times 2\sqrt{37}$$

Multiply 3 and 2 to get 6.

$$2\sqrt{3}+6\sqrt{37}$$