Solve 2/3-sqrt{7}-2/3+sqrt{3}

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

Evaluate

$$\frac{\sqrt{3}}{3}+\sqrt{7}+2\approx 5.22310158$$

Solution Steps

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

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

Consider \left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 3. Square \sqrt{7}.

$$\frac{2\left(3+\sqrt{7}\right)}{9-7}-\frac{2}{3+\sqrt{3}}$$

Subtract 7 from 9 to get 2.

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

Cancel out 2 and 2.

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

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

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

Consider \left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 3. Square \sqrt{3}.

$$3+\sqrt{7}-\frac{2\left(3-\sqrt{3}\right)}{9-3}$$

Subtract 3 from 9 to get 6.

$$3+\sqrt{7}-\frac{2\left(3-\sqrt{3}\right)}{6}$$

Divide 2\left(3-\sqrt{3}\right) by 6 to get \frac{1}{3}\left(3-\sqrt{3}\right).

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

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

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

Cancel out 3 and 3.

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

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

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

To find the opposite of 1-\frac{1}{3}\sqrt{3}, find the opposite of each term.

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

The opposite of -\frac{1}{3}\sqrt{3} is \frac{1}{3}\sqrt{3}.

$$3+\sqrt{7}-1+\frac{1}{3}\sqrt{3}$$

Subtract 1 from 3 to get 2.

$$2+\sqrt{7}+\frac{1}{3}\sqrt{3}$$

Factor

$$\frac{\sqrt{3} + 3 \sqrt{7} + 6}{3} = 5.223101580254217$$