Solve frac{2sqrt{3}+1}{(sqrt{3}+1)^2+1}

$$\frac { 2 \sqrt { 3 } + 1 } { ( \sqrt { 3 } + 1 ) ^ { 2 } + 1 }$$

Evaluate

$$\frac{8\sqrt{3}-7}{13}\approx 0.527415882$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3}+1\right)^{2}.

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

The square of \sqrt{3} is 3.

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

Add 3 and 1 to get 4.

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

Add 4 and 1 to get 5.

$$\frac{2\sqrt{3}+1}{5+2\sqrt{3}}$$

Rationalize the denominator of \frac{2\sqrt{3}+1}{5+2\sqrt{3}} by multiplying numerator and denominator by 5-2\sqrt{3}.

$$\frac{\left(2\sqrt{3}+1\right)\left(5-2\sqrt{3}\right)}{\left(5+2\sqrt{3}\right)\left(5-2\sqrt{3}\right)}$$

Consider \left(5+2\sqrt{3}\right)\left(5-2\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}.

$$\frac{\left(2\sqrt{3}+1\right)\left(5-2\sqrt{3}\right)}{5^{2}-\left(2\sqrt{3}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(2\sqrt{3}+1\right)\left(5-2\sqrt{3}\right)}{25-\left(2\sqrt{3}\right)^{2}}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\frac{\left(2\sqrt{3}+1\right)\left(5-2\sqrt{3}\right)}{25-2^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(2\sqrt{3}+1\right)\left(5-2\sqrt{3}\right)}{25-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(2\sqrt{3}+1\right)\left(5-2\sqrt{3}\right)}{25-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(2\sqrt{3}+1\right)\left(5-2\sqrt{3}\right)}{25-12}$$

Subtract 12 from 25 to get 13.

$$\frac{\left(2\sqrt{3}+1\right)\left(5-2\sqrt{3}\right)}{13}$$

Use the distributive property to multiply 2\sqrt{3}+1 by 5-2\sqrt{3} and combine like terms.

$$\frac{8\sqrt{3}-4\left(\sqrt{3}\right)^{2}+5}{13}$$

The square of \sqrt{3} is 3.

$$\frac{8\sqrt{3}-4\times 3+5}{13}$$

Multiply -4 and 3 to get -12.

$$\frac{8\sqrt{3}-12+5}{13}$$

Add -12 and 5 to get -7.

$$\frac{8\sqrt{3}-7}{13}$$