Solve 12/sqrt{2+sqrt{3}}-1/sqrt{3}

$$\frac { 12 } { \sqrt { 2 } + \sqrt { 3 } } - \frac { 1 } { \sqrt { 3 } }$$

Evaluate

$$\frac{35\sqrt{3}}{3}-12\sqrt{2}\approx 3.236696673$$

Solution Steps

Rationalize the denominator of \frac{12}{\sqrt{2}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{3}.

$$\frac{12\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}-\frac{1}{\sqrt{3}}$$

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

$$\frac{12\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}-\frac{1}{\sqrt{3}}$$

Square \sqrt{2}. Square \sqrt{3}.

$$\frac{12\left(\sqrt{2}-\sqrt{3}\right)}{2-3}-\frac{1}{\sqrt{3}}$$

Subtract 3 from 2 to get -1.

$$\frac{12\left(\sqrt{2}-\sqrt{3}\right)}{-1}-\frac{1}{\sqrt{3}}$$

Anything divided by -1 gives its opposite.

$$-12\left(\sqrt{2}-\sqrt{3}\right)-\frac{1}{\sqrt{3}}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$-12\left(\sqrt{2}-\sqrt{3}\right)-\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$-12\left(\sqrt{2}-\sqrt{3}\right)-\frac{\sqrt{3}}{3}$$

Use the distributive property to multiply -12 by \sqrt{2}-\sqrt{3}.

$$-12\sqrt{2}+12\sqrt{3}-\frac{\sqrt{3}}{3}$$

Combine 12\sqrt{3} and -\frac{\sqrt{3}}{3} to get \frac{35}{3}\sqrt{3}.

$$-12\sqrt{2}+\frac{35}{3}\sqrt{3}$$