Solve =20-10(sqrt{frac{2.20/10}-1)^2}{2}

$$= 20 - \frac { 10 ( \sqrt { \frac { 2.20 } { 10 } } - 1 ) ^ { 2 } } { 2 }$$

Evaluate

$$\sqrt{22}+13.9\approx 18.59041576$$

Short Solution Steps

Expand \frac{2.2}{10} by multiplying both numerator and the denominator by 10.

$$20-\frac{10\left(\sqrt{\frac{22}{100}}-1\right)^{2}}{2}$$

Reduce the fraction \frac{22}{100} to lowest terms by extracting and canceling out 2.

$$20-\frac{10\left(\sqrt{\frac{11}{50}}-1\right)^{2}}{2}$$

Rewrite the square root of the division \sqrt{\frac{11}{50}} as the division of square roots \frac{\sqrt{11}}{\sqrt{50}}.

$$20-\frac{10\left(\frac{\sqrt{11}}{\sqrt{50}}-1\right)^{2}}{2}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$20-\frac{10\left(\frac{\sqrt{11}}{5\sqrt{2}}-1\right)^{2}}{2}$$

Rationalize the denominator of \frac{\sqrt{11}}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$20-\frac{10\left(\frac{\sqrt{11}\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}-1\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$20-\frac{10\left(\frac{\sqrt{11}\sqrt{2}}{5\times 2}-1\right)^{2}}{2}$$

To multiply \sqrt{11} and \sqrt{2}, multiply the numbers under the square root.

$$20-\frac{10\left(\frac{\sqrt{22}}{5\times 2}-1\right)^{2}}{2}$$

Multiply 5 and 2 to get 10.

$$20-\frac{10\left(\frac{\sqrt{22}}{10}-1\right)^{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{10}{10}.

$$20-\frac{10\left(\frac{\sqrt{22}}{10}-\frac{10}{10}\right)^{2}}{2}$$

Since \frac{\sqrt{22}}{10} and \frac{10}{10} have the same denominator, subtract them by subtracting their numerators.

$$20-\frac{10\times \left(\frac{\sqrt{22}-10}{10}\right)^{2}}{2}$$

To raise \frac{\sqrt{22}-10}{10} to a power, raise both numerator and denominator to the power and then divide.

$$20-\frac{10\times \frac{\left(\sqrt{22}-10\right)^{2}}{10^{2}}}{2}$$

Express 10\times \frac{\left(\sqrt{22}-10\right)^{2}}{10^{2}} as a single fraction.

$$20-\frac{\frac{10\left(\sqrt{22}-10\right)^{2}}{10^{2}}}{2}$$

Cancel out 10 in both numerator and denominator.

$$20-\frac{\frac{\left(\sqrt{22}-10\right)^{2}}{10}}{2}$$

Express \frac{\frac{\left(\sqrt{22}-10\right)^{2}}{10}}{2} as a single fraction.

$$20-\frac{\left(\sqrt{22}-10\right)^{2}}{10\times 2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{22}-10\right)^{2}.

$$20-\frac{\left(\sqrt{22}\right)^{2}-20\sqrt{22}+100}{10\times 2}$$

The square of \sqrt{22} is 22.

$$20-\frac{22-20\sqrt{22}+100}{10\times 2}$$

Add 22 and 100 to get 122.

$$20-\frac{122-20\sqrt{22}}{10\times 2}$$

Multiply 10 and 2 to get 20.

$$20-\frac{122-20\sqrt{22}}{20}$$

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

$$\frac{20\times 20}{20}-\frac{122-20\sqrt{22}}{20}$$

Since \frac{20\times 20}{20} and \frac{122-20\sqrt{22}}{20} have the same denominator, subtract them by subtracting their numerators.

$$\frac{20\times 20-\left(122-20\sqrt{22}\right)}{20}$$

Do the multiplications in 20\times 20-\left(122-20\sqrt{22}\right).

$$\frac{400-122+20\sqrt{22}}{20}$$

Do the calculations in 400-122+20\sqrt{22}.

$$\frac{278+20\sqrt{22}}{20}$$

Expand

$$\sqrt{22} + 13.9 = 18.59041576$$

Short Solution Steps

Expand \frac{2.2}{10} by multiplying both numerator and the denominator by 10.

$$20-\frac{10\left(\sqrt{\frac{22}{100}}-1\right)^{2}}{2}$$

Reduce the fraction \frac{22}{100} to lowest terms by extracting and canceling out 2.

$$20-\frac{10\left(\sqrt{\frac{11}{50}}-1\right)^{2}}{2}$$

Rewrite the square root of the division \sqrt{\frac{11}{50}} as the division of square roots \frac{\sqrt{11}}{\sqrt{50}}.

$$20-\frac{10\left(\frac{\sqrt{11}}{\sqrt{50}}-1\right)^{2}}{2}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$20-\frac{10\left(\frac{\sqrt{11}}{5\sqrt{2}}-1\right)^{2}}{2}$$

Rationalize the denominator of \frac{\sqrt{11}}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$20-\frac{10\left(\frac{\sqrt{11}\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}-1\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$20-\frac{10\left(\frac{\sqrt{11}\sqrt{2}}{5\times 2}-1\right)^{2}}{2}$$

To multiply \sqrt{11} and \sqrt{2}, multiply the numbers under the square root.

$$20-\frac{10\left(\frac{\sqrt{22}}{5\times 2}-1\right)^{2}}{2}$$

Multiply 5 and 2 to get 10.

$$20-\frac{10\left(\frac{\sqrt{22}}{10}-1\right)^{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{10}{10}.

$$20-\frac{10\left(\frac{\sqrt{22}}{10}-\frac{10}{10}\right)^{2}}{2}$$

Since \frac{\sqrt{22}}{10} and \frac{10}{10} have the same denominator, subtract them by subtracting their numerators.

$$20-\frac{10\times \left(\frac{\sqrt{22}-10}{10}\right)^{2}}{2}$$

To raise \frac{\sqrt{22}-10}{10} to a power, raise both numerator and denominator to the power and then divide.

$$20-\frac{10\times \frac{\left(\sqrt{22}-10\right)^{2}}{10^{2}}}{2}$$

Express 10\times \frac{\left(\sqrt{22}-10\right)^{2}}{10^{2}} as a single fraction.

$$20-\frac{\frac{10\left(\sqrt{22}-10\right)^{2}}{10^{2}}}{2}$$

Cancel out 10 in both numerator and denominator.

$$20-\frac{\frac{\left(\sqrt{22}-10\right)^{2}}{10}}{2}$$

Express \frac{\frac{\left(\sqrt{22}-10\right)^{2}}{10}}{2} as a single fraction.

$$20-\frac{\left(\sqrt{22}-10\right)^{2}}{10\times 2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{22}-10\right)^{2}.

$$20-\frac{\left(\sqrt{22}\right)^{2}-20\sqrt{22}+100}{10\times 2}$$

The square of \sqrt{22} is 22.

$$20-\frac{22-20\sqrt{22}+100}{10\times 2}$$

Add 22 and 100 to get 122.

$$20-\frac{122-20\sqrt{22}}{10\times 2}$$

Multiply 10 and 2 to get 20.

$$20-\frac{122-20\sqrt{22}}{20}$$

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

$$\frac{20\times 20}{20}-\frac{122-20\sqrt{22}}{20}$$

Since \frac{20\times 20}{20} and \frac{122-20\sqrt{22}}{20} have the same denominator, subtract them by subtracting their numerators.

$$\frac{20\times 20-\left(122-20\sqrt{22}\right)}{20}$$

Do the multiplications in 20\times 20-\left(122-20\sqrt{22}\right).

$$\frac{400-122+20\sqrt{22}}{20}$$

Do the calculations in 400-122+20\sqrt{22}.

$$\frac{278+20\sqrt{22}}{20}$$