Solve 1/2+sqrt{3}+sqrt{27}

$$\frac { 1 } { 2 + \sqrt { 3 } } + \sqrt { 27 }$$

Evaluate

$$2\left(\sqrt{3}+1\right)\approx 5.464101615$$

Solution Steps

Rationalize the denominator of \frac{1}{2+\sqrt{3}} by multiplying numerator and denominator by 2-\sqrt{3}.

$$\frac{2-\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}+\sqrt{27}$$

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

$$\frac{2-\sqrt{3}}{2^{2}-\left(\sqrt{3}\right)^{2}}+\sqrt{27}$$

Square 2. Square \sqrt{3}.

$$\frac{2-\sqrt{3}}{4-3}+\sqrt{27}$$

Subtract 3 from 4 to get 1.

$$\frac{2-\sqrt{3}}{1}+\sqrt{27}$$

Anything divided by one gives itself.

$$2-\sqrt{3}+\sqrt{27}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

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

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

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