Solve sqrt{3}(2sqrt{7}-3)-(2+sqrt{7})divsqrt{3}+sqrt{27}

$$\sqrt { 3 } ( 2 \sqrt { 7 } - 3 ) - ( 2 + \sqrt { 7 } ) \div \sqrt { 3 } + \sqrt { 27 }$$

Evaluate

$$\frac{5\sqrt{21}-2\sqrt{3}}{3}\approx 6.48292562$$

Short Solution Steps

Rationalize the denominator of \frac{2+\sqrt{7}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\sqrt{3}\left(2\sqrt{7}-3\right)-\frac{\left(2+\sqrt{7}\right)\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{27}$$

The square of \sqrt{3} is 3.

$$\sqrt{3}\left(2\sqrt{7}-3\right)-\frac{\left(2+\sqrt{7}\right)\sqrt{3}}{3}+\sqrt{27}$$

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}.

$$\sqrt{3}\left(2\sqrt{7}-3\right)-\frac{\left(2+\sqrt{7}\right)\sqrt{3}}{3}+3\sqrt{3}$$

Use the distributive property to multiply \sqrt{3} by 2\sqrt{7}-3.

$$2\sqrt{3}\sqrt{7}-3\sqrt{3}-\frac{\left(2+\sqrt{7}\right)\sqrt{3}}{3}+3\sqrt{3}$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$2\sqrt{21}-3\sqrt{3}-\frac{\left(2+\sqrt{7}\right)\sqrt{3}}{3}+3\sqrt{3}$$

Use the distributive property to multiply 2+\sqrt{7} by \sqrt{3}.

$$2\sqrt{21}-3\sqrt{3}-\frac{2\sqrt{3}+\sqrt{7}\sqrt{3}}{3}+3\sqrt{3}$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$2\sqrt{21}-3\sqrt{3}-\frac{2\sqrt{3}+\sqrt{21}}{3}+3\sqrt{3}$$

Combine -3\sqrt{3} and 3\sqrt{3} to get 0.

$$2\sqrt{21}-\frac{2\sqrt{3}+\sqrt{21}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{21} times \frac{3}{3}.

$$\frac{3\times 2\sqrt{21}}{3}-\frac{2\sqrt{3}+\sqrt{21}}{3}$$

Since \frac{3\times 2\sqrt{21}}{3} and \frac{2\sqrt{3}+\sqrt{21}}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 2\sqrt{21}-\left(2\sqrt{3}+\sqrt{21}\right)}{3}$$

Do the multiplications in 3\times 2\sqrt{21}-\left(2\sqrt{3}+\sqrt{21}\right).

$$\frac{6\sqrt{21}-2\sqrt{3}-\sqrt{21}}{3}$$

Do the calculations in 6\sqrt{21}-2\sqrt{3}-\sqrt{21}.

$$\frac{5\sqrt{21}-2\sqrt{3}}{3}$$