Solve 4/7+sqrt{3}

$$\frac { 4 } { 7 + \sqrt { 3 } }$$

Evaluate

$$\frac{14-2\sqrt{3}}{23}\approx 0.458082538$$

Solution Steps

Rationalize the denominator of \frac{4}{7+\sqrt{3}} by multiplying numerator and denominator by 7-\sqrt{3}.

$$\frac{4\left(7-\sqrt{3}\right)}{\left(7+\sqrt{3}\right)\left(7-\sqrt{3}\right)}$$

Consider \left(7+\sqrt{3}\right)\left(7-\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}.

$$\frac{4\left(7-\sqrt{3}\right)}{7^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 7. Square \sqrt{3}.

$$\frac{4\left(7-\sqrt{3}\right)}{49-3}$$

Subtract 3 from 49 to get 46.

$$\frac{4\left(7-\sqrt{3}\right)}{46}$$

Divide 4\left(7-\sqrt{3}\right) by 46 to get \frac{2}{23}\left(7-\sqrt{3}\right).

$$\frac{2}{23}\left(7-\sqrt{3}\right)$$

Use the distributive property to multiply \frac{2}{23} by 7-\sqrt{3}.

$$\frac{2}{23}\times 7+\frac{2}{23}\left(-1\right)\sqrt{3}$$

Express \frac{2}{23}\times 7 as a single fraction.

$$\frac{2\times 7}{23}+\frac{2}{23}\left(-1\right)\sqrt{3}$$

Multiply 2 and 7 to get 14.

$$\frac{14}{23}+\frac{2}{23}\left(-1\right)\sqrt{3}$$

Multiply \frac{2}{23} and -1 to get -\frac{2}{23}.

$$\frac{14}{23}-\frac{2}{23}\sqrt{3}$$