Solve 100/(sqrt{3/2+.5/2)}

$$\frac{ 100 }{ ( \frac{ \sqrt{ 3 } }{ 2 } + \frac{ .5 }{ 2 } ) }$$

Evaluate

$$\frac{800\sqrt{3}-400}{11}\approx 89.603695096$$

Short Solution Steps

Expand \frac{0.5}{2} by multiplying both numerator and the denominator by 10.

$$\frac{100}{\frac{\sqrt{3}}{2}+\frac{5}{20}}$$

Reduce the fraction \frac{5}{20} to lowest terms by extracting and canceling out 5.

$$\frac{100}{\frac{\sqrt{3}}{2}+\frac{1}{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{\sqrt{3}}{2} times \frac{2}{2}.

$$\frac{100}{\frac{2\sqrt{3}}{4}+\frac{1}{4}}$$

Since \frac{2\sqrt{3}}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{100}{\frac{2\sqrt{3}+1}{4}}$$

Divide 100 by \frac{2\sqrt{3}+1}{4} by multiplying 100 by the reciprocal of \frac{2\sqrt{3}+1}{4}.

$$\frac{100\times 4}{2\sqrt{3}+1}$$

Rationalize the denominator of \frac{100\times 4}{2\sqrt{3}+1} by multiplying numerator and denominator by 2\sqrt{3}-1.

$$\frac{100\times 4\left(2\sqrt{3}-1\right)}{\left(2\sqrt{3}+1\right)\left(2\sqrt{3}-1\right)}$$

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}.

$$\frac{100\times 4\left(2\sqrt{3}-1\right)}{\left(2\sqrt{3}\right)^{2}-1^{2}}$$

Multiply 100 and 4 to get 400.

$$\frac{400\left(2\sqrt{3}-1\right)}{\left(2\sqrt{3}\right)^{2}-1^{2}}$$

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

$$\frac{400\left(2\sqrt{3}-1\right)}{2^{2}\left(\sqrt{3}\right)^{2}-1^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{400\left(2\sqrt{3}-1\right)}{4\left(\sqrt{3}\right)^{2}-1^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{400\left(2\sqrt{3}-1\right)}{4\times 3-1^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{400\left(2\sqrt{3}-1\right)}{12-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{400\left(2\sqrt{3}-1\right)}{12-1}$$

Subtract 1 from 12 to get 11.

$$\frac{400\left(2\sqrt{3}-1\right)}{11}$$

Use the distributive property to multiply 400 by 2\sqrt{3}-1.

$$\frac{800\sqrt{3}-400}{11}$$