Solve frac{12+7sqrt{3}}{sqrt{3}+2}

$$\frac { 12 + 7 \sqrt { 3 } } { \sqrt { 3 } + 2 }$$

Evaluate

$$2\sqrt{3}+3\approx 6.464101615$$

Solution Steps

Rationalize the denominator of \frac{12+7\sqrt{3}}{\sqrt{3}+2} by multiplying numerator and denominator by \sqrt{3}-2.

$$\frac{\left(12+7\sqrt{3}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{3}+2\right)\left(\sqrt{3}-2\right)}$$

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}.

$$\frac{\left(12+7\sqrt{3}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{3}\right)^{2}-2^{2}}$$

Square \sqrt{3}. Square 2.

$$\frac{\left(12+7\sqrt{3}\right)\left(\sqrt{3}-2\right)}{3-4}$$

Subtract 4 from 3 to get -1.

$$\frac{\left(12+7\sqrt{3}\right)\left(\sqrt{3}-2\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\left(12+7\sqrt{3}\right)\left(\sqrt{3}-2\right)$$

Apply the distributive property by multiplying each term of 12+7\sqrt{3} by each term of \sqrt{3}-2.

$$-\left(12\sqrt{3}-24+7\left(\sqrt{3}\right)^{2}-14\sqrt{3}\right)$$

The square of \sqrt{3} is 3.

$$-\left(12\sqrt{3}-24+7\times 3-14\sqrt{3}\right)$$

Multiply 7 and 3 to get 21.

$$-\left(12\sqrt{3}-24+21-14\sqrt{3}\right)$$

Add -24 and 21 to get -3.

$$-\left(12\sqrt{3}-3-14\sqrt{3}\right)$$

Combine 12\sqrt{3} and -14\sqrt{3} to get -2\sqrt{3}.

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

To find the opposite of -2\sqrt{3}-3, find the opposite of each term.

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

The opposite of -2\sqrt{3} is 2\sqrt{3}.

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

The opposite of -3 is 3.

$$2\sqrt{3}+3$$