Solve sqrt{27}-1/3sqrt{18}-sqrt{16/3}

$$\sqrt { 27 } - \frac { 1 } { 3 } \sqrt { 18 } - \sqrt { \frac { 16 } { 3 } }$$

Evaluate

$$\frac{5\sqrt{3}}{3}-\sqrt{2}\approx 1.472537784$$

Short Solution Steps

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}.

$$3\sqrt{3}-\frac{1}{3}\sqrt{18}-\sqrt{\frac{16}{3}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$3\sqrt{3}-\frac{1}{3}\times 3\sqrt{2}-\sqrt{\frac{16}{3}}$$

Cancel out 3 and 3.

$$3\sqrt{3}-\sqrt{2}-\sqrt{\frac{16}{3}}$$

Rewrite the square root of the division \sqrt{\frac{16}{3}} as the division of square roots \frac{\sqrt{16}}{\sqrt{3}}.

$$3\sqrt{3}-\sqrt{2}-\frac{\sqrt{16}}{\sqrt{3}}$$

Calculate the square root of 16 and get 4.

$$3\sqrt{3}-\sqrt{2}-\frac{4}{\sqrt{3}}$$

Rationalize the denominator of \frac{4}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

$$3\sqrt{3}-\sqrt{2}-\frac{4\sqrt{3}}{3}$$

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

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

Since \frac{3\left(3\sqrt{3}-\sqrt{2}\right)}{3} and \frac{4\sqrt{3}}{3} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Do the calculations in 9\sqrt{3}-3\sqrt{2}-4\sqrt{3}.

$$\frac{5\sqrt{3}-3\sqrt{2}}{3}$$

Factor

$$\frac{5 \sqrt{3} - 3 \sqrt{2}}{3} = 1.4725377835750333$$