Solve frac{{left(frac{9-9sqrt{5}}{4}right)}^2}{9}*5-5

$$\frac{ { \left( \frac{ 9-9 \sqrt{ 5 } }{ 4 } \right) }^{ 2 } }{ 9 } \times 5-5$$

Evaluate

$$\frac{95-45\sqrt{5}}{8}\approx -0.702882373$$

Short Solution Steps

To raise \frac{9-9\sqrt{5}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$5\times \frac{\frac{\left(9-9\sqrt{5}\right)^{2}}{4^{2}}}{9}-5$$

Express \frac{\frac{\left(9-9\sqrt{5}\right)^{2}}{4^{2}}}{9} as a single fraction.

$$5\times \frac{\left(9-9\sqrt{5}\right)^{2}}{4^{2}\times 9}-5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9-9\sqrt{5}\right)^{2}.

$$5\times \frac{81-162\sqrt{5}+81\left(\sqrt{5}\right)^{2}}{4^{2}\times 9}-5$$

The square of \sqrt{5} is 5.

$$5\times \frac{81-162\sqrt{5}+81\times 5}{4^{2}\times 9}-5$$

Multiply 81 and 5 to get 405.

$$5\times \frac{81-162\sqrt{5}+405}{4^{2}\times 9}-5$$

Add 81 and 405 to get 486.

$$5\times \frac{486-162\sqrt{5}}{4^{2}\times 9}-5$$

Calculate 4 to the power of 2 and get 16.

$$5\times \frac{486-162\sqrt{5}}{16\times 9}-5$$

Multiply 16 and 9 to get 144.

$$5\times \frac{486-162\sqrt{5}}{144}-5$$

Express 5\times \frac{486-162\sqrt{5}}{144} as a single fraction.

$$\frac{5\left(486-162\sqrt{5}\right)}{144}-5$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{144}{144}.

$$\frac{5\left(486-162\sqrt{5}\right)}{144}-\frac{5\times 144}{144}$$

Since \frac{5\left(486-162\sqrt{5}\right)}{144} and \frac{5\times 144}{144} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(486-162\sqrt{5}\right)-5\times 144}{144}$$

Do the multiplications in 5\left(486-162\sqrt{5}\right)-5\times 144.

$$\frac{2430-810\sqrt{5}-720}{144}$$

Do the calculations in 2430-810\sqrt{5}-720.

$$\frac{1710-810\sqrt{5}}{144}$$

Expand

$$\frac{95 - 45 \sqrt{5}}{8} = -0.7028823734363172$$

Short Solution Steps

To raise \frac{9-9\sqrt{5}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$5\times \frac{\frac{\left(9-9\sqrt{5}\right)^{2}}{4^{2}}}{9}-5$$

Express \frac{\frac{\left(9-9\sqrt{5}\right)^{2}}{4^{2}}}{9} as a single fraction.

$$5\times \frac{\left(9-9\sqrt{5}\right)^{2}}{4^{2}\times 9}-5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9-9\sqrt{5}\right)^{2}.

$$5\times \frac{81-162\sqrt{5}+81\left(\sqrt{5}\right)^{2}}{4^{2}\times 9}-5$$

The square of \sqrt{5} is 5.

$$5\times \frac{81-162\sqrt{5}+81\times 5}{4^{2}\times 9}-5$$

Multiply 81 and 5 to get 405.

$$5\times \frac{81-162\sqrt{5}+405}{4^{2}\times 9}-5$$

Add 81 and 405 to get 486.

$$5\times \frac{486-162\sqrt{5}}{4^{2}\times 9}-5$$

Calculate 4 to the power of 2 and get 16.

$$5\times \frac{486-162\sqrt{5}}{16\times 9}-5$$

Multiply 16 and 9 to get 144.

$$5\times \frac{486-162\sqrt{5}}{144}-5$$

Express 5\times \frac{486-162\sqrt{5}}{144} as a single fraction.

$$\frac{5\left(486-162\sqrt{5}\right)}{144}-5$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{144}{144}.

$$\frac{5\left(486-162\sqrt{5}\right)}{144}-\frac{5\times 144}{144}$$

Since \frac{5\left(486-162\sqrt{5}\right)}{144} and \frac{5\times 144}{144} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(486-162\sqrt{5}\right)-5\times 144}{144}$$

Do the multiplications in 5\left(486-162\sqrt{5}\right)-5\times 144.

$$\frac{2430-810\sqrt{5}-720}{144}$$

Do the calculations in 2430-810\sqrt{5}-720.

$$\frac{1710-810\sqrt{5}}{144}$$