Solve sqrt{[(21/2-1/6+0.2)*9]-11/4}=

$$\sqrt { [ ( 2 \frac { 1 } { 2 } - \frac { 1 } { 6 } + 0.2 ) \cdot 9 ] - \frac { 11 } { 4 } } =$$

Evaluate

$$\frac{\sqrt{2005}}{10}\approx 4.477722635$$

Solution Steps

Multiply 2 and 2 to get 4.

$$\sqrt{\left(\frac{4+1}{2}-\frac{1}{6}+0.2\right)\times 9-\frac{11}{4}}$$

Add 4 and 1 to get 5.

$$\sqrt{\left(\frac{5}{2}-\frac{1}{6}+0.2\right)\times 9-\frac{11}{4}}$$

Least common multiple of 2 and 6 is 6. Convert \frac{5}{2} and \frac{1}{6} to fractions with denominator 6.

$$\sqrt{\left(\frac{15}{6}-\frac{1}{6}+0.2\right)\times 9-\frac{11}{4}}$$

Since \frac{15}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\left(\frac{15-1}{6}+0.2\right)\times 9-\frac{11}{4}}$$

Subtract 1 from 15 to get 14.

$$\sqrt{\left(\frac{14}{6}+0.2\right)\times 9-\frac{11}{4}}$$

Reduce the fraction \frac{14}{6} to lowest terms by extracting and canceling out 2.

$$\sqrt{\left(\frac{7}{3}+0.2\right)\times 9-\frac{11}{4}}$$

Convert decimal number 0.2 to fraction \frac{2}{10}. Reduce the fraction \frac{2}{10} to lowest terms by extracting and canceling out 2.

$$\sqrt{\left(\frac{7}{3}+\frac{1}{5}\right)\times 9-\frac{11}{4}}$$

Least common multiple of 3 and 5 is 15. Convert \frac{7}{3} and \frac{1}{5} to fractions with denominator 15.

$$\sqrt{\left(\frac{35}{15}+\frac{3}{15}\right)\times 9-\frac{11}{4}}$$

Since \frac{35}{15} and \frac{3}{15} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{35+3}{15}\times 9-\frac{11}{4}}$$

Add 35 and 3 to get 38.

$$\sqrt{\frac{38}{15}\times 9-\frac{11}{4}}$$

Express \frac{38}{15}\times 9 as a single fraction.

$$\sqrt{\frac{38\times 9}{15}-\frac{11}{4}}$$

Multiply 38 and 9 to get 342.

$$\sqrt{\frac{342}{15}-\frac{11}{4}}$$

Reduce the fraction \frac{342}{15} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{114}{5}-\frac{11}{4}}$$

Least common multiple of 5 and 4 is 20. Convert \frac{114}{5} and \frac{11}{4} to fractions with denominator 20.

$$\sqrt{\frac{456}{20}-\frac{55}{20}}$$

Since \frac{456}{20} and \frac{55}{20} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{456-55}{20}}$$

Subtract 55 from 456 to get 401.

$$\sqrt{\frac{401}{20}}$$

Rewrite the square root of the division \sqrt{\frac{401}{20}} as the division of square roots \frac{\sqrt{401}}{\sqrt{20}}.

$$\frac{\sqrt{401}}{\sqrt{20}}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{\sqrt{401}}{2\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{401}}{2\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{401}\sqrt{5}}{2\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{401}\sqrt{5}}{2\times 5}$$

To multiply \sqrt{401} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{2005}}{2\times 5}$$

Multiply 2 and 5 to get 10.

$$\frac{\sqrt{2005}}{10}$$