Solve -4/7+sqrt{3}

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

Evaluate

$$\frac{2\left(\sqrt{3}-7\right)}{23}\approx -0.458082538$$

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.

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

Fraction \frac{-14}{23} can be rewritten as -\frac{14}{23} by extracting the negative sign.

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

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

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