Solve 3/sqrt{5+2}*8

$$\frac { 3 } { \sqrt { 5 } + 2 } \cdot 8$$

Evaluate

$$24\left(\sqrt{5}-2\right)\approx 5.66563146$$

Solution Steps

Rationalize the denominator of \frac{3}{\sqrt{5}+2} by multiplying numerator and denominator by \sqrt{5}-2.

$$8\times \frac{3\left(\sqrt{5}-2\right)}{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}$$

Consider \left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$8\times \frac{3\left(\sqrt{5}-2\right)}{\left(\sqrt{5}\right)^{2}-2^{2}}$$

Square \sqrt{5}. Square 2.

$$8\times \frac{3\left(\sqrt{5}-2\right)}{5-4}$$

Subtract 4 from 5 to get 1.

$$8\times \frac{3\left(\sqrt{5}-2\right)}{1}$$

Anything divided by one gives itself.

$$8\times 3\left(\sqrt{5}-2\right)$$

Use the distributive property to multiply 3 by \sqrt{5}-2.

$$8\left(3\sqrt{5}-6\right)$$

Use the distributive property to multiply 8 by 3\sqrt{5}-6.

$$24\sqrt{5}-48$$