Solve frac{frac{sqrt{3}}{2}-1.54/3.08^2}{frac{sqrt{3}}{2}+1.54/3.08^2}

$$\frac{ \frac{ \sqrt{ 3 } }{ 2 } - \frac{ 1.54 }{ { 3.08 }^{ 2 } } }{ \frac{ \sqrt{ 3 } }{ 2 } + \frac{ 1.54 }{ { 3.08 }^{ 2 } } }$$

Evaluate

$$\frac{9206-1925\sqrt{3}}{8581}\approx 0.684279477$$

Short Solution Steps

Calculate 3.08 to the power of 2 and get 9.4864.

$$\frac{\frac{\sqrt{3}}{2}-\frac{1.54}{9.4864}}{\frac{\sqrt{3}}{2}+\frac{1.54}{3.08^{2}}}$$

Expand \frac{1.54}{9.4864} by multiplying both numerator and the denominator by 10000.

$$\frac{\frac{\sqrt{3}}{2}-\frac{15400}{94864}}{\frac{\sqrt{3}}{2}+\frac{1.54}{3.08^{2}}}$$

Reduce the fraction \frac{15400}{94864} to lowest terms by extracting and canceling out 616.

$$\frac{\frac{\sqrt{3}}{2}-\frac{25}{154}}{\frac{\sqrt{3}}{2}+\frac{1.54}{3.08^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 154 is 154. Multiply \frac{\sqrt{3}}{2} times \frac{77}{77}.

$$\frac{\frac{77\sqrt{3}}{154}-\frac{25}{154}}{\frac{\sqrt{3}}{2}+\frac{1.54}{3.08^{2}}}$$

Since \frac{77\sqrt{3}}{154} and \frac{25}{154} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{77\sqrt{3}-25}{154}}{\frac{\sqrt{3}}{2}+\frac{1.54}{3.08^{2}}}$$

Calculate 3.08 to the power of 2 and get 9.4864.

$$\frac{\frac{77\sqrt{3}-25}{154}}{\frac{\sqrt{3}}{2}+\frac{1.54}{9.4864}}$$

Expand \frac{1.54}{9.4864} by multiplying both numerator and the denominator by 10000.

$$\frac{\frac{77\sqrt{3}-25}{154}}{\frac{\sqrt{3}}{2}+\frac{15400}{94864}}$$

Reduce the fraction \frac{15400}{94864} to lowest terms by extracting and canceling out 616.

$$\frac{\frac{77\sqrt{3}-25}{154}}{\frac{\sqrt{3}}{2}+\frac{25}{154}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 154 is 154. Multiply \frac{\sqrt{3}}{2} times \frac{77}{77}.

$$\frac{\frac{77\sqrt{3}-25}{154}}{\frac{77\sqrt{3}}{154}+\frac{25}{154}}$$

Since \frac{77\sqrt{3}}{154} and \frac{25}{154} have the same denominator, add them by adding their numerators.

$$\frac{\frac{77\sqrt{3}-25}{154}}{\frac{77\sqrt{3}+25}{154}}$$

Divide \frac{77\sqrt{3}-25}{154} by \frac{77\sqrt{3}+25}{154} by multiplying \frac{77\sqrt{3}-25}{154} by the reciprocal of \frac{77\sqrt{3}+25}{154}.

$$\frac{\left(77\sqrt{3}-25\right)\times 154}{154\left(77\sqrt{3}+25\right)}$$

Cancel out 154 in both numerator and denominator.

$$\frac{77\sqrt{3}-25}{77\sqrt{3}+25}$$

Rationalize the denominator of \frac{77\sqrt{3}-25}{77\sqrt{3}+25} by multiplying numerator and denominator by 77\sqrt{3}-25.

$$\frac{\left(77\sqrt{3}-25\right)\left(77\sqrt{3}-25\right)}{\left(77\sqrt{3}+25\right)\left(77\sqrt{3}-25\right)}$$

Consider \left(77\sqrt{3}+25\right)\left(77\sqrt{3}-25\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(77\sqrt{3}-25\right)\left(77\sqrt{3}-25\right)}{\left(77\sqrt{3}\right)^{2}-25^{2}}$$

Multiply 77\sqrt{3}-25 and 77\sqrt{3}-25 to get \left(77\sqrt{3}-25\right)^{2}.

$$\frac{\left(77\sqrt{3}-25\right)^{2}}{\left(77\sqrt{3}\right)^{2}-25^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(77\sqrt{3}-25\right)^{2}.

$$\frac{5929\left(\sqrt{3}\right)^{2}-3850\sqrt{3}+625}{\left(77\sqrt{3}\right)^{2}-25^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5929\times 3-3850\sqrt{3}+625}{\left(77\sqrt{3}\right)^{2}-25^{2}}$$

Multiply 5929 and 3 to get 17787.

$$\frac{17787-3850\sqrt{3}+625}{\left(77\sqrt{3}\right)^{2}-25^{2}}$$

Add 17787 and 625 to get 18412.

$$\frac{18412-3850\sqrt{3}}{\left(77\sqrt{3}\right)^{2}-25^{2}}$$

Expand \left(77\sqrt{3}\right)^{2}.

$$\frac{18412-3850\sqrt{3}}{77^{2}\left(\sqrt{3}\right)^{2}-25^{2}}$$

Calculate 77 to the power of 2 and get 5929.

$$\frac{18412-3850\sqrt{3}}{5929\left(\sqrt{3}\right)^{2}-25^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{18412-3850\sqrt{3}}{5929\times 3-25^{2}}$$

Multiply 5929 and 3 to get 17787.

$$\frac{18412-3850\sqrt{3}}{17787-25^{2}}$$

Calculate 25 to the power of 2 and get 625.

$$\frac{18412-3850\sqrt{3}}{17787-625}$$

Subtract 625 from 17787 to get 17162.

$$\frac{18412-3850\sqrt{3}}{17162}$$

Factor

$$\frac{9206 - 1925 \sqrt{3}}{8581} = 0.6842794773837444$$