Solve 5/2*(81.07-1)*(-1+sqrt{1+4*150^2*8.854*10^-12*left(81.07-1right)^2/9.81*5*1000}

$$\frac{ 5 }{ 2 \times (81.07-1) } \times (-1+ \sqrt{ 1+ \frac{ 4 \times { 150 }^{ 2 } \times 8.854 \times { 10 }^{ -12 } \times { \left(81.07-1 \right) }^{ 2 } }{ 9.81 \times 5 \times 1000 } }$$

Evaluate

$$\frac{\sqrt{2970250309368204606070}}{1745526000000}-\frac{250}{8007}\approx 1.626008626 \cdot 10^{-9}$$

Solution Steps

Subtract 1 from 81.07 to get 80.07.

$$\frac{5}{2\times 80.07}\left(-1+\sqrt{1+\frac{4\times 150^{2}\times 8.854\times 10^{-12}\left(81.07-1\right)^{2}}{9.81\times 5\times 1000}}\right)$$

Multiply 2 and 80.07 to get 160.14.

$$\frac{5}{160.14}\left(-1+\sqrt{1+\frac{4\times 150^{2}\times 8.854\times 10^{-12}\left(81.07-1\right)^{2}}{9.81\times 5\times 1000}}\right)$$

Expand \frac{5}{160.14} by multiplying both numerator and the denominator by 100.

$$\frac{500}{16014}\left(-1+\sqrt{1+\frac{4\times 150^{2}\times 8.854\times 10^{-12}\left(81.07-1\right)^{2}}{9.81\times 5\times 1000}}\right)$$

Reduce the fraction \frac{500}{16014} to lowest terms by extracting and canceling out 2.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{4\times 150^{2}\times 8.854\times 10^{-12}\left(81.07-1\right)^{2}}{9.81\times 5\times 1000}}\right)$$

Cancel out 4 in both numerator and denominator.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{8.854\times 10^{-12}\times 150^{2}\left(81.07-1\right)^{2}}{5\times 9.81\times 250}}\right)$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{8.854\times \frac{1}{1000000000000}\times 150^{2}\left(81.07-1\right)^{2}}{5\times 9.81\times 250}}\right)$$

Multiply 8.854 and \frac{1}{1000000000000} to get \frac{4427}{500000000000000}.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{\frac{4427}{500000000000000}\times 150^{2}\left(81.07-1\right)^{2}}{5\times 9.81\times 250}}\right)$$

Calculate 150 to the power of 2 and get 22500.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{\frac{4427}{500000000000000}\times 22500\left(81.07-1\right)^{2}}{5\times 9.81\times 250}}\right)$$

Multiply \frac{4427}{500000000000000} and 22500 to get \frac{39843}{200000000000}.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{\frac{39843}{200000000000}\left(81.07-1\right)^{2}}{5\times 9.81\times 250}}\right)$$

Subtract 1 from 81.07 to get 80.07.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{\frac{39843}{200000000000}\times 80.07^{2}}{5\times 9.81\times 250}}\right)$$

Calculate 80.07 to the power of 2 and get 6411.2049.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{\frac{39843}{200000000000}\times 6411.2049}{5\times 9.81\times 250}}\right)$$

Multiply \frac{39843}{200000000000} and 6411.2049 to get \frac{2554416368307}{2000000000000000}.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{\frac{2554416368307}{2000000000000000}}{5\times 9.81\times 250}}\right)$$

Multiply 5 and 9.81 to get 49.05.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{\frac{2554416368307}{2000000000000000}}{49.05\times 250}}\right)$$

Multiply 49.05 and 250 to get 12262.5.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{\frac{2554416368307}{2000000000000000}}{12262.5}}\right)$$

Express \frac{\frac{2554416368307}{2000000000000000}}{12262.5} as a single fraction.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{2554416368307}{2000000000000000\times 12262.5}}\right)$$

Multiply 2000000000000000 and 12262.5 to get 24525000000000000000.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{2554416368307}{24525000000000000000}}\right)$$

Reduce the fraction \frac{2554416368307}{24525000000000000000} to lowest terms by extracting and canceling out 9.

$$\frac{250}{8007}\left(-1+\sqrt{1+\frac{283824040923}{2725000000000000000}}\right)$$

Add 1 and \frac{283824040923}{2725000000000000000} to get \frac{2725000283824040923}{2725000000000000000}.

$$\frac{250}{8007}\left(-1+\sqrt{\frac{2725000283824040923}{2725000000000000000}}\right)$$

Rewrite the square root of the division \sqrt{\frac{2725000283824040923}{2725000000000000000}} as the division of square roots \frac{\sqrt{2725000283824040923}}{\sqrt{2725000000000000000}}.

$$\frac{250}{8007}\left(-1+\frac{\sqrt{2725000283824040923}}{\sqrt{2725000000000000000}}\right)$$

Factor 2725000000000000000=50000000^{2}\times 1090. Rewrite the square root of the product \sqrt{50000000^{2}\times 1090} as the product of square roots \sqrt{50000000^{2}}\sqrt{1090}. Take the square root of 50000000^{2}.

$$\frac{250}{8007}\left(-1+\frac{\sqrt{2725000283824040923}}{50000000\sqrt{1090}}\right)$$

Rationalize the denominator of \frac{\sqrt{2725000283824040923}}{50000000\sqrt{1090}} by multiplying numerator and denominator by \sqrt{1090}.

$$\frac{250}{8007}\left(-1+\frac{\sqrt{2725000283824040923}\sqrt{1090}}{50000000\left(\sqrt{1090}\right)^{2}}\right)$$

The square of \sqrt{1090} is 1090.

$$\frac{250}{8007}\left(-1+\frac{\sqrt{2725000283824040923}\sqrt{1090}}{50000000\times 1090}\right)$$

To multiply \sqrt{2725000283824040923} and \sqrt{1090}, multiply the numbers under the square root.

$$\frac{250}{8007}\left(-1+\frac{\sqrt{2970250309368204606070}}{50000000\times 1090}\right)$$

Multiply 50000000 and 1090 to get 54500000000.

$$\frac{250}{8007}\left(-1+\frac{\sqrt{2970250309368204606070}}{54500000000}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -1 times \frac{54500000000}{54500000000}.

$$\frac{250}{8007}\left(-\frac{54500000000}{54500000000}+\frac{\sqrt{2970250309368204606070}}{54500000000}\right)$$

Since -\frac{54500000000}{54500000000} and \frac{\sqrt{2970250309368204606070}}{54500000000} have the same denominator, add them by adding their numerators.

$$\frac{250}{8007}\times \frac{-54500000000+\sqrt{2970250309368204606070}}{54500000000}$$

Multiply \frac{250}{8007} times \frac{-54500000000+\sqrt{2970250309368204606070}}{54500000000} by multiplying numerator times numerator and denominator times denominator.

$$\frac{250\left(-54500000000+\sqrt{2970250309368204606070}\right)}{8007\times 54500000000}$$

Cancel out 250 in both numerator and denominator.

$$\frac{\sqrt{2970250309368204606070}-54500000000}{8007\times 218000000}$$

Multiply 8007 and 218000000 to get 1745526000000.

$$\frac{\sqrt{2970250309368204606070}-54500000000}{1745526000000}$$