Solve (3)2sqrt{8}-4sqrt{1/27}+3sqrt{48}

$$( 3 ) 2 \sqrt { 8 } - 4 \sqrt { \frac { 1 } { 27 } } + 3 \sqrt { 48 }$$

Evaluate

$$\frac{104\sqrt{3}}{9}+12\sqrt{2}\approx 36.98537208$$

Short Solution Steps

Multiply 2 and 3 to get 6.

$$6\sqrt{8}-4\sqrt{\frac{1}{27}}+3\sqrt{48}$$

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}.

$$6\times 2\sqrt{2}-4\sqrt{\frac{1}{27}}+3\sqrt{48}$$

Multiply 6 and 2 to get 12.

$$12\sqrt{2}-4\sqrt{\frac{1}{27}}+3\sqrt{48}$$

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

$$12\sqrt{2}-4\times \frac{\sqrt{1}}{\sqrt{27}}+3\sqrt{48}$$

Calculate the square root of 1 and get 1.

$$12\sqrt{2}-4\times \frac{1}{\sqrt{27}}+3\sqrt{48}$$

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}.

$$12\sqrt{2}-4\times \frac{1}{3\sqrt{3}}+3\sqrt{48}$$

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

$$12\sqrt{2}-4\times \frac{\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}+3\sqrt{48}$$

The square of \sqrt{3} is 3.

$$12\sqrt{2}-4\times \frac{\sqrt{3}}{3\times 3}+3\sqrt{48}$$

Multiply 3 and 3 to get 9.

$$12\sqrt{2}-4\times \frac{\sqrt{3}}{9}+3\sqrt{48}$$

Express -4\times \frac{\sqrt{3}}{9} as a single fraction.

$$12\sqrt{2}+\frac{-4\sqrt{3}}{9}+3\sqrt{48}$$

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}.

$$12\sqrt{2}+\frac{-4\sqrt{3}}{9}+3\times 4\sqrt{3}$$

Multiply 3 and 4 to get 12.

$$12\sqrt{2}+\frac{-4\sqrt{3}}{9}+12\sqrt{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12\sqrt{2}+12\sqrt{3} times \frac{9}{9}.

$$\frac{9\left(12\sqrt{2}+12\sqrt{3}\right)}{9}+\frac{-4\sqrt{3}}{9}$$

Since \frac{9\left(12\sqrt{2}+12\sqrt{3}\right)}{9} and \frac{-4\sqrt{3}}{9} have the same denominator, add them by adding their numerators.

$$\frac{9\left(12\sqrt{2}+12\sqrt{3}\right)-4\sqrt{3}}{9}$$

Do the multiplications in 9\left(12\sqrt{2}+12\sqrt{3}\right)-4\sqrt{3}.

$$\frac{108\sqrt{2}+108\sqrt{3}-4\sqrt{3}}{9}$$

Do the calculations in 108\sqrt{2}+108\sqrt{3}-4\sqrt{3}.

$$\frac{108\sqrt{2}+104\sqrt{3}}{9}$$