Solve 12/2+sqrt{13}

$$\frac{ 12 }{ 2+ \sqrt{ 13 } }$$

Evaluate

$$\frac{4\left(\sqrt{13}-2\right)}{3}\approx 2.140735034$$

Solution Steps

Rationalize the denominator of \frac{12}{2+\sqrt{13}} by multiplying numerator and denominator by 2-\sqrt{13}.

$$\frac{12\left(2-\sqrt{13}\right)}{\left(2+\sqrt{13}\right)\left(2-\sqrt{13}\right)}$$

Consider \left(2+\sqrt{13}\right)\left(2-\sqrt{13}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{12\left(2-\sqrt{13}\right)}{2^{2}-\left(\sqrt{13}\right)^{2}}$$

Square 2. Square \sqrt{13}.

$$\frac{12\left(2-\sqrt{13}\right)}{4-13}$$

Subtract 13 from 4 to get -9.

$$\frac{12\left(2-\sqrt{13}\right)}{-9}$$

Divide 12\left(2-\sqrt{13}\right) by -9 to get -\frac{4}{3}\left(2-\sqrt{13}\right).

$$-\frac{4}{3}\left(2-\sqrt{13}\right)$$

Use the distributive property to multiply -\frac{4}{3} by 2-\sqrt{13}.

$$-\frac{4}{3}\times 2-\frac{4}{3}\left(-1\right)\sqrt{13}$$

Express -\frac{4}{3}\times 2 as a single fraction.

$$\frac{-4\times 2}{3}-\frac{4}{3}\left(-1\right)\sqrt{13}$$

Multiply -4 and 2 to get -8.

$$\frac{-8}{3}-\frac{4}{3}\left(-1\right)\sqrt{13}$$

Fraction \frac{-8}{3} can be rewritten as -\frac{8}{3} by extracting the negative sign.

$$-\frac{8}{3}-\frac{4}{3}\left(-1\right)\sqrt{13}$$

Multiply -\frac{4}{3} and -1 to get \frac{4}{3}.

$$-\frac{8}{3}+\frac{4}{3}\sqrt{13}$$