Solve 9sqrt{3}+7sqrt{12}-5sqrt{48}+2sqrt{1/3}

$$9 \sqrt { 3 } + 7 \sqrt { 12 } - 5 \sqrt { 48 } + 2 \sqrt { \frac { 1 } { 3 } }$$

Evaluate

$$\frac{11\sqrt{3}}{3}\approx 6.350852961$$

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}.

$$9\sqrt{3}+7\times 2\sqrt{3}-5\sqrt{48}+2\sqrt{\frac{1}{3}}$$

Multiply 7 and 2 to get 14.

$$9\sqrt{3}+14\sqrt{3}-5\sqrt{48}+2\sqrt{\frac{1}{3}}$$

Combine 9\sqrt{3} and 14\sqrt{3} to get 23\sqrt{3}.

$$23\sqrt{3}-5\sqrt{48}+2\sqrt{\frac{1}{3}}$$

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}.

$$23\sqrt{3}-5\times 4\sqrt{3}+2\sqrt{\frac{1}{3}}$$

Multiply -5 and 4 to get -20.

$$23\sqrt{3}-20\sqrt{3}+2\sqrt{\frac{1}{3}}$$

Combine 23\sqrt{3} and -20\sqrt{3} to get 3\sqrt{3}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{3} is 3.

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

Express 2\times \frac{\sqrt{3}}{3} as a single fraction.

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

Combine 3\sqrt{3} and \frac{2\sqrt{3}}{3} to get \frac{11}{3}\sqrt{3}.

$$\frac{11}{3}\sqrt{3}$$