Solve 2+5/4-5/9/2sqrt{2-frac{sqrt{5}}{2}}

$$\frac { 2 + \frac { 5 } { 4 } - \frac { 5 } { 9 } } { 2 \sqrt { 2 } - \frac { \sqrt { 5 } } { 2 } }$$

Evaluate

$$\frac{97\left(\sqrt{5}+4\sqrt{2}\right)}{486}\approx 1.575336329$$

Short Solution Steps

Convert 2 to fraction \frac{8}{4}.

$$\frac{\frac{8}{4}+\frac{5}{4}-\frac{5}{9}}{2\sqrt{2}-\frac{\sqrt{5}}{2}}$$

Since \frac{8}{4} and \frac{5}{4} have the same denominator, add them by adding their numerators.

$$\frac{\frac{8+5}{4}-\frac{5}{9}}{2\sqrt{2}-\frac{\sqrt{5}}{2}}$$

Add 8 and 5 to get 13.

$$\frac{\frac{13}{4}-\frac{5}{9}}{2\sqrt{2}-\frac{\sqrt{5}}{2}}$$

Least common multiple of 4 and 9 is 36. Convert \frac{13}{4} and \frac{5}{9} to fractions with denominator 36.

$$\frac{\frac{117}{36}-\frac{20}{36}}{2\sqrt{2}-\frac{\sqrt{5}}{2}}$$

Since \frac{117}{36} and \frac{20}{36} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{117-20}{36}}{2\sqrt{2}-\frac{\sqrt{5}}{2}}$$

Subtract 20 from 117 to get 97.

$$\frac{\frac{97}{36}}{2\sqrt{2}-\frac{\sqrt{5}}{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{2} times \frac{2}{2}.

$$\frac{\frac{97}{36}}{\frac{2\times 2\sqrt{2}}{2}-\frac{\sqrt{5}}{2}}$$

Since \frac{2\times 2\sqrt{2}}{2} and \frac{\sqrt{5}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{97}{36}}{\frac{2\times 2\sqrt{2}-\sqrt{5}}{2}}$$

Do the multiplications in 2\times 2\sqrt{2}-\sqrt{5}.

$$\frac{\frac{97}{36}}{\frac{4\sqrt{2}-\sqrt{5}}{2}}$$

Divide \frac{97}{36} by \frac{4\sqrt{2}-\sqrt{5}}{2} by multiplying \frac{97}{36} by the reciprocal of \frac{4\sqrt{2}-\sqrt{5}}{2}.

$$\frac{97\times 2}{36\left(4\sqrt{2}-\sqrt{5}\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{97}{18\left(-\sqrt{5}+4\sqrt{2}\right)}$$

Use the distributive property to multiply 18 by -\sqrt{5}+4\sqrt{2}.

$$\frac{97}{-18\sqrt{5}+72\sqrt{2}}$$

Rationalize the denominator of \frac{97}{-18\sqrt{5}+72\sqrt{2}} by multiplying numerator and denominator by -18\sqrt{5}-72\sqrt{2}.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{\left(-18\sqrt{5}+72\sqrt{2}\right)\left(-18\sqrt{5}-72\sqrt{2}\right)}$$

Consider \left(-18\sqrt{5}+72\sqrt{2}\right)\left(-18\sqrt{5}-72\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{\left(-18\sqrt{5}\right)^{2}-\left(72\sqrt{2}\right)^{2}}$$

Expand \left(-18\sqrt{5}\right)^{2}.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{\left(-18\right)^{2}\left(\sqrt{5}\right)^{2}-\left(72\sqrt{2}\right)^{2}}$$

Calculate -18 to the power of 2 and get 324.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{324\left(\sqrt{5}\right)^{2}-\left(72\sqrt{2}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{324\times 5-\left(72\sqrt{2}\right)^{2}}$$

Multiply 324 and 5 to get 1620.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{1620-\left(72\sqrt{2}\right)^{2}}$$

Expand \left(72\sqrt{2}\right)^{2}.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{1620-72^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 72 to the power of 2 and get 5184.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{1620-5184\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{1620-5184\times 2}$$

Multiply 5184 and 2 to get 10368.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{1620-10368}$$

Subtract 10368 from 1620 to get -8748.

$$\frac{97\left(-18\sqrt{5}-72\sqrt{2}\right)}{-8748}$$

Use the distributive property to multiply 97 by -18\sqrt{5}-72\sqrt{2}.

$$\frac{-1746\sqrt{5}-6984\sqrt{2}}{-8748}$$

Factor

$$\frac{97 {(\sqrt{5} + 4 \sqrt{2})}}{486} = 1.575336329255639$$