Solve 1/2(690/234)^2(180/345)[-1+sqrt{1+(2*234*345/620+180)}]

$$\frac { 1 } { 2 } ( \frac { 690 } { 234 } ) ^ { 2 } ( \frac { 180 } { 345 } ) [ - 1 + \sqrt { 1 + ( \frac { 2 \cdot 234 \cdot 345 } { 620 + 180 } ) } ]$$

Evaluate

$$\frac{115\left(\sqrt{81130}-20\right)}{1014}\approx 30.035333063$$

Solution Steps

Reduce the fraction \frac{690}{234} to lowest terms by extracting and canceling out 6.

$$\frac{1}{2}\times \left(\frac{115}{39}\right)^{2}\times \frac{180}{345}\left(-1+\sqrt{1+\frac{2\times 234\times 345}{620+180}}\right)$$

Calculate \frac{115}{39} to the power of 2 and get \frac{13225}{1521}.

$$\frac{1}{2}\times \frac{13225}{1521}\times \frac{180}{345}\left(-1+\sqrt{1+\frac{2\times 234\times 345}{620+180}}\right)$$

Multiply \frac{1}{2} and \frac{13225}{1521} to get \frac{13225}{3042}.

$$\frac{13225}{3042}\times \frac{180}{345}\left(-1+\sqrt{1+\frac{2\times 234\times 345}{620+180}}\right)$$

Reduce the fraction \frac{180}{345} to lowest terms by extracting and canceling out 15.

$$\frac{13225}{3042}\times \frac{12}{23}\left(-1+\sqrt{1+\frac{2\times 234\times 345}{620+180}}\right)$$

Multiply \frac{13225}{3042} and \frac{12}{23} to get \frac{1150}{507}.

$$\frac{1150}{507}\left(-1+\sqrt{1+\frac{2\times 234\times 345}{620+180}}\right)$$

Multiply 2 and 234 to get 468.

$$\frac{1150}{507}\left(-1+\sqrt{1+\frac{468\times 345}{620+180}}\right)$$

Multiply 468 and 345 to get 161460.

$$\frac{1150}{507}\left(-1+\sqrt{1+\frac{161460}{620+180}}\right)$$

Add 620 and 180 to get 800.

$$\frac{1150}{507}\left(-1+\sqrt{1+\frac{161460}{800}}\right)$$

Reduce the fraction \frac{161460}{800} to lowest terms by extracting and canceling out 20.

$$\frac{1150}{507}\left(-1+\sqrt{1+\frac{8073}{40}}\right)$$

Add 1 and \frac{8073}{40} to get \frac{8113}{40}.

$$\frac{1150}{507}\left(-1+\sqrt{\frac{8113}{40}}\right)$$

Rewrite the square root of the division \sqrt{\frac{8113}{40}} as the division of square roots \frac{\sqrt{8113}}{\sqrt{40}}.

$$\frac{1150}{507}\left(-1+\frac{\sqrt{8113}}{\sqrt{40}}\right)$$

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

$$\frac{1150}{507}\left(-1+\frac{\sqrt{8113}}{2\sqrt{10}}\right)$$

Rationalize the denominator of \frac{\sqrt{8113}}{2\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{1150}{507}\left(-1+\frac{\sqrt{8113}\sqrt{10}}{2\left(\sqrt{10}\right)^{2}}\right)$$

The square of \sqrt{10} is 10.

$$\frac{1150}{507}\left(-1+\frac{\sqrt{8113}\sqrt{10}}{2\times 10}\right)$$

To multiply \sqrt{8113} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{1150}{507}\left(-1+\frac{\sqrt{81130}}{2\times 10}\right)$$

Multiply 2 and 10 to get 20.

$$\frac{1150}{507}\left(-1+\frac{\sqrt{81130}}{20}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -1 times \frac{20}{20}.

$$\frac{1150}{507}\left(-\frac{20}{20}+\frac{\sqrt{81130}}{20}\right)$$

Since -\frac{20}{20} and \frac{\sqrt{81130}}{20} have the same denominator, add them by adding their numerators.

$$\frac{1150}{507}\times \frac{-20+\sqrt{81130}}{20}$$

Multiply \frac{1150}{507} times \frac{-20+\sqrt{81130}}{20} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1150\left(-20+\sqrt{81130}\right)}{507\times 20}$$

Cancel out 10 in both numerator and denominator.

$$\frac{115\left(\sqrt{81130}-20\right)}{2\times 507}$$

Multiply 2 and 507 to get 1014.

$$\frac{115\left(\sqrt{81130}-20\right)}{1014}$$

Use the distributive property to multiply 115 by \sqrt{81130}-20.

$$\frac{115\sqrt{81130}-2300}{1014}$$