Solve 3/2-sqrt{3}+4/sqrt{3+1}

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

Evaluate

$$5\sqrt{3}+4\approx 12.660254038$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

Rationalize the denominator of \frac{4}{\sqrt{3}+1} by multiplying numerator and denominator by \sqrt{3}-1.

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

Consider \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Divide 4\left(\sqrt{3}-1\right) by 2 to get 2\left(\sqrt{3}-1\right).

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

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

$$6+3\sqrt{3}+2\left(\sqrt{3}-1\right)$$

Use the distributive property to multiply 2 by \sqrt{3}-1.

$$6+3\sqrt{3}+2\sqrt{3}-2$$

Combine 3\sqrt{3} and 2\sqrt{3} to get 5\sqrt{3}.

$$6+5\sqrt{3}-2$$

Subtract 2 from 6 to get 4.

$$4+5\sqrt{3}$$