Solve 4/sqrt{3+2}*frac{sqrt{3}}{sqrt{3}}

$$\frac{ 4 }{ \sqrt{ 3 } +2 } \times \frac{ \sqrt{ 3 } }{ \sqrt{ 3 } }$$

Evaluate

$$8-4\sqrt{3}\approx 1.07179677$$

Solution Steps

Divide \sqrt{3} by \sqrt{3} to get 1.

$$\frac{4}{\sqrt{3}+2}\times 1$$

Rationalize the denominator of \frac{4}{\sqrt{3}+2} by multiplying numerator and denominator by \sqrt{3}-2.

$$\frac{4\left(\sqrt{3}-2\right)}{\left(\sqrt{3}+2\right)\left(\sqrt{3}-2\right)}\times 1$$

Consider \left(\sqrt{3}+2\right)\left(\sqrt{3}-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}.

$$\frac{4\left(\sqrt{3}-2\right)}{\left(\sqrt{3}\right)^{2}-2^{2}}\times 1$$

Square \sqrt{3}. Square 2.

$$\frac{4\left(\sqrt{3}-2\right)}{3-4}\times 1$$

Subtract 4 from 3 to get -1.

$$\frac{4\left(\sqrt{3}-2\right)}{-1}\times 1$$

Anything divided by -1 gives its opposite.

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

Use the distributive property to multiply -4 by \sqrt{3}-2.

$$\left(-4\sqrt{3}+8\right)\times 1$$

Use the distributive property to multiply -4\sqrt{3}+8 by 1.

$$-4\sqrt{3}+8$$