Solve 2/sqrt{3}-sqrt{14/9}divsqrt{2/21}-(1-sqrt{3})*(-sqrt{3}

$$\frac { 2 } { \sqrt { 3 } } - \sqrt { \frac { 14 } { 9 } } \div \sqrt { \frac { 2 } { 21 } } - ( 1 - \sqrt { 3 } ) \times ( - \sqrt { 3 }$$

Evaluate

$$-\frac{2\sqrt{3}}{3}-3\approx -4.154700538$$

Solution Steps

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

$$\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\frac{\sqrt{\frac{14}{9}}}{\sqrt{\frac{2}{21}}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}}{3}-\frac{\sqrt{\frac{14}{9}}}{\sqrt{\frac{2}{21}}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Rewrite the square root of the division \sqrt{\frac{14}{9}} as the division of square roots \frac{\sqrt{14}}{\sqrt{9}}.

$$\frac{2\sqrt{3}}{3}-\frac{\frac{\sqrt{14}}{\sqrt{9}}}{\sqrt{\frac{2}{21}}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Calculate the square root of 9 and get 3.

$$\frac{2\sqrt{3}}{3}-\frac{\frac{\sqrt{14}}{3}}{\sqrt{\frac{2}{21}}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Rewrite the square root of the division \sqrt{\frac{2}{21}} as the division of square roots \frac{\sqrt{2}}{\sqrt{21}}.

$$\frac{2\sqrt{3}}{3}-\frac{\frac{\sqrt{14}}{3}}{\frac{\sqrt{2}}{\sqrt{21}}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{2\sqrt{3}}{3}-\frac{\frac{\sqrt{14}}{3}}{\frac{\sqrt{2}\sqrt{21}}{\left(\sqrt{21}\right)^{2}}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

The square of \sqrt{21} is 21.

$$\frac{2\sqrt{3}}{3}-\frac{\frac{\sqrt{14}}{3}}{\frac{\sqrt{2}\sqrt{21}}{21}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

To multiply \sqrt{2} and \sqrt{21}, multiply the numbers under the square root.

$$\frac{2\sqrt{3}}{3}-\frac{\frac{\sqrt{14}}{3}}{\frac{\sqrt{42}}{21}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Divide \frac{\sqrt{14}}{3} by \frac{\sqrt{42}}{21} by multiplying \frac{\sqrt{14}}{3} by the reciprocal of \frac{\sqrt{42}}{21}.

$$\frac{2\sqrt{3}}{3}-\frac{\sqrt{14}\times 21}{3\sqrt{42}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Cancel out 3 in both numerator and denominator.

$$\frac{2\sqrt{3}}{3}-\frac{7\sqrt{14}}{\sqrt{42}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Rationalize the denominator of \frac{7\sqrt{14}}{\sqrt{42}} by multiplying numerator and denominator by \sqrt{42}.

$$\frac{2\sqrt{3}}{3}-\frac{7\sqrt{14}\sqrt{42}}{\left(\sqrt{42}\right)^{2}}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

The square of \sqrt{42} is 42.

$$\frac{2\sqrt{3}}{3}-\frac{7\sqrt{14}\sqrt{42}}{42}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Factor 42=14\times 3. Rewrite the square root of the product \sqrt{14\times 3} as the product of square roots \sqrt{14}\sqrt{3}.

$$\frac{2\sqrt{3}}{3}-\frac{7\sqrt{14}\sqrt{14}\sqrt{3}}{42}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Multiply \sqrt{14} and \sqrt{14} to get 14.

$$\frac{2\sqrt{3}}{3}-\frac{7\times 14\sqrt{3}}{42}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Multiply 7 and 14 to get 98.

$$\frac{2\sqrt{3}}{3}-\frac{98\sqrt{3}}{42}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Divide 98\sqrt{3} by 42 to get \frac{7}{3}\sqrt{3}.

$$\frac{2\sqrt{3}}{3}-\frac{7}{3}\sqrt{3}-\left(1-\sqrt{3}\right)\left(-\sqrt{3}\right)$$

Use the distributive property to multiply 1-\sqrt{3} by -\sqrt{3}.

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

Multiply -1 and -1 to get 1.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

To find the opposite of -\sqrt{3}+3, find the opposite of each term.

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

Combine \frac{2\sqrt{3}}{3} and -\frac{7}{3}\sqrt{3} to get -\frac{5}{3}\sqrt{3}.

$$-\frac{5}{3}\sqrt{3}-\left(-\sqrt{3}\right)-3$$

Multiply -1 and -1 to get 1.

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

Combine -\frac{5}{3}\sqrt{3} and \sqrt{3} to get -\frac{2}{3}\sqrt{3}.

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