Solve 8/sqrt{7+3}

$$\frac{ 8 }{ \sqrt{ 7 } +3 }$$

Evaluate

$$12-4\sqrt{7}\approx 1.416994756$$

Solution Steps

Rationalize the denominator of \frac{8}{\sqrt{7}+3} by multiplying numerator and denominator by \sqrt{7}-3.

$$\frac{8\left(\sqrt{7}-3\right)}{\left(\sqrt{7}+3\right)\left(\sqrt{7}-3\right)}$$

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

$$\frac{8\left(\sqrt{7}-3\right)}{\left(\sqrt{7}\right)^{2}-3^{2}}$$

Square \sqrt{7}. Square 3.

$$\frac{8\left(\sqrt{7}-3\right)}{7-9}$$

Subtract 9 from 7 to get -2.

$$\frac{8\left(\sqrt{7}-3\right)}{-2}$$

Divide 8\left(\sqrt{7}-3\right) by -2 to get -4\left(\sqrt{7}-3\right).

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

Use the distributive property to multiply -4 by \sqrt{7}-3.

$$-4\sqrt{7}+12$$