Solve 17/2sqrt{3+1}

$$\frac { 17 } { 2 \sqrt { 3 } + 1 }$$

Evaluate

$$\frac{34\sqrt{3}-17}{11}\approx 3.808157042$$

Solution Steps

Rationalize the denominator of \frac{17}{2\sqrt{3}+1} by multiplying numerator and denominator by 2\sqrt{3}-1.

$$\frac{17\left(2\sqrt{3}-1\right)}{\left(2\sqrt{3}+1\right)\left(2\sqrt{3}-1\right)}$$

Consider \left(2\sqrt{3}+1\right)\left(2\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}.

$$\frac{17\left(2\sqrt{3}-1\right)}{\left(2\sqrt{3}\right)^{2}-1^{2}}$$

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

$$\frac{17\left(2\sqrt{3}-1\right)}{2^{2}\left(\sqrt{3}\right)^{2}-1^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{17\left(2\sqrt{3}-1\right)}{4\left(\sqrt{3}\right)^{2}-1^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{17\left(2\sqrt{3}-1\right)}{4\times 3-1^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{17\left(2\sqrt{3}-1\right)}{12-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{17\left(2\sqrt{3}-1\right)}{12-1}$$

Subtract 1 from 12 to get 11.

$$\frac{17\left(2\sqrt{3}-1\right)}{11}$$

Use the distributive property to multiply 17 by 2\sqrt{3}-1.

$$\frac{34\sqrt{3}-17}{11}$$