Solve frac{frac{sqrt{3}}{2}-154/308^2}{frac{sqrt{3}}{2}+154/308^2}

$$\frac{ \frac{ \sqrt{ 3 } }{ 2 } - \frac{ 154 }{ { 308 }^{ 2 } } }{ \frac{ \sqrt{ 3 } }{ 2 } + \frac{ 154 }{ { 308 }^{ 2 } } }$$

Evaluate

$$\frac{284593-616\sqrt{3}}{284591}\approx 0.996257987$$

Short Solution Steps

Calculate 308 to the power of 2 and get 94864.

$$\frac{\frac{\sqrt{3}}{2}-\frac{154}{94864}}{\frac{\sqrt{3}}{2}+\frac{154}{308^{2}}}$$

Reduce the fraction \frac{154}{94864} to lowest terms by extracting and canceling out 154.

$$\frac{\frac{\sqrt{3}}{2}-\frac{1}{616}}{\frac{\sqrt{3}}{2}+\frac{154}{308^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 616 is 616. Multiply \frac{\sqrt{3}}{2} times \frac{308}{308}.

$$\frac{\frac{308\sqrt{3}}{616}-\frac{1}{616}}{\frac{\sqrt{3}}{2}+\frac{154}{308^{2}}}$$

Since \frac{308\sqrt{3}}{616} and \frac{1}{616} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{308\sqrt{3}-1}{616}}{\frac{\sqrt{3}}{2}+\frac{154}{308^{2}}}$$

Calculate 308 to the power of 2 and get 94864.

$$\frac{\frac{308\sqrt{3}-1}{616}}{\frac{\sqrt{3}}{2}+\frac{154}{94864}}$$

Reduce the fraction \frac{154}{94864} to lowest terms by extracting and canceling out 154.

$$\frac{\frac{308\sqrt{3}-1}{616}}{\frac{\sqrt{3}}{2}+\frac{1}{616}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 616 is 616. Multiply \frac{\sqrt{3}}{2} times \frac{308}{308}.

$$\frac{\frac{308\sqrt{3}-1}{616}}{\frac{308\sqrt{3}}{616}+\frac{1}{616}}$$

Since \frac{308\sqrt{3}}{616} and \frac{1}{616} have the same denominator, add them by adding their numerators.

$$\frac{\frac{308\sqrt{3}-1}{616}}{\frac{308\sqrt{3}+1}{616}}$$

Divide \frac{308\sqrt{3}-1}{616} by \frac{308\sqrt{3}+1}{616} by multiplying \frac{308\sqrt{3}-1}{616} by the reciprocal of \frac{308\sqrt{3}+1}{616}.

$$\frac{\left(308\sqrt{3}-1\right)\times 616}{616\left(308\sqrt{3}+1\right)}$$

Cancel out 616 in both numerator and denominator.

$$\frac{308\sqrt{3}-1}{308\sqrt{3}+1}$$

Rationalize the denominator of \frac{308\sqrt{3}-1}{308\sqrt{3}+1} by multiplying numerator and denominator by 308\sqrt{3}-1.

$$\frac{\left(308\sqrt{3}-1\right)\left(308\sqrt{3}-1\right)}{\left(308\sqrt{3}+1\right)\left(308\sqrt{3}-1\right)}$$

Consider \left(308\sqrt{3}+1\right)\left(308\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}.

$$\frac{\left(308\sqrt{3}-1\right)\left(308\sqrt{3}-1\right)}{\left(308\sqrt{3}\right)^{2}-1^{2}}$$

Multiply 308\sqrt{3}-1 and 308\sqrt{3}-1 to get \left(308\sqrt{3}-1\right)^{2}.

$$\frac{\left(308\sqrt{3}-1\right)^{2}}{\left(308\sqrt{3}\right)^{2}-1^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(308\sqrt{3}-1\right)^{2}.

$$\frac{94864\left(\sqrt{3}\right)^{2}-616\sqrt{3}+1}{\left(308\sqrt{3}\right)^{2}-1^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{94864\times 3-616\sqrt{3}+1}{\left(308\sqrt{3}\right)^{2}-1^{2}}$$

Multiply 94864 and 3 to get 284592.

$$\frac{284592-616\sqrt{3}+1}{\left(308\sqrt{3}\right)^{2}-1^{2}}$$

Add 284592 and 1 to get 284593.

$$\frac{284593-616\sqrt{3}}{\left(308\sqrt{3}\right)^{2}-1^{2}}$$

Expand \left(308\sqrt{3}\right)^{2}.

$$\frac{284593-616\sqrt{3}}{308^{2}\left(\sqrt{3}\right)^{2}-1^{2}}$$

Calculate 308 to the power of 2 and get 94864.

$$\frac{284593-616\sqrt{3}}{94864\left(\sqrt{3}\right)^{2}-1^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{284593-616\sqrt{3}}{94864\times 3-1^{2}}$$

Multiply 94864 and 3 to get 284592.

$$\frac{284593-616\sqrt{3}}{284592-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{284593-616\sqrt{3}}{284592-1}$$

Subtract 1 from 284592 to get 284591.

$$\frac{284593-616\sqrt{3}}{284591}$$

Factor

$$\frac{284593 - 616 \sqrt{3}}{284591} = 0.9962579867337251$$