Solve sqrt{0.1(-14*5%)^2+0.3(-2.5%)^2+0.4(2.5%)^2+0.2(5.5%)^2}

$$\sqrt { 0.1 ( - 14 \cdot 5 \% ) ^ { 2 } + 0.3 ( - 2.5 \% ) ^ { 2 } + 0.4 ( 2.5 \% ) ^ { 2 } + 0.2 ( 5.5 \% ) ^ { 2 } }$$

Evaluate

$$\frac{\sqrt{200170}}{2000}\approx 0.22370181$$

Solution Steps

Reduce the fraction \frac{5}{100} to lowest terms by extracting and canceling out 5.

$$\sqrt{0.1\left(-14\times \frac{1}{20}\right)^{2}+0.3\left(-\frac{2.5}{100}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Multiply -14 and \frac{1}{20} to get -\frac{7}{10}.

$$\sqrt{0.1\left(-\frac{7}{10}\right)^{2}+0.3\left(-\frac{2.5}{100}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Calculate -\frac{7}{10} to the power of 2 and get \frac{49}{100}.

$$\sqrt{0.1\times \frac{49}{100}+0.3\left(-\frac{2.5}{100}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Multiply 0.1 and \frac{49}{100} to get \frac{49}{1000}.

$$\sqrt{\frac{49}{1000}+0.3\left(-\frac{2.5}{100}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Expand \frac{2.5}{100} by multiplying both numerator and the denominator by 10.

$$\sqrt{\frac{49}{1000}+0.3\left(-\frac{25}{1000}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Reduce the fraction \frac{25}{1000} to lowest terms by extracting and canceling out 25.

$$\sqrt{\frac{49}{1000}+0.3\left(-\frac{1}{40}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Calculate -\frac{1}{40} to the power of 2 and get \frac{1}{1600}.

$$\sqrt{\frac{49}{1000}+0.3\times \frac{1}{1600}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Multiply 0.3 and \frac{1}{1600} to get \frac{3}{16000}.

$$\sqrt{\frac{49}{1000}+\frac{3}{16000}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Add \frac{49}{1000} and \frac{3}{16000} to get \frac{787}{16000}.

$$\sqrt{\frac{787}{16000}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Expand \frac{2.5}{100} by multiplying both numerator and the denominator by 10.

$$\sqrt{\frac{787}{16000}+0.4\times \left(\frac{25}{1000}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Reduce the fraction \frac{25}{1000} to lowest terms by extracting and canceling out 25.

$$\sqrt{\frac{787}{16000}+0.4\times \left(\frac{1}{40}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Calculate \frac{1}{40} to the power of 2 and get \frac{1}{1600}.

$$\sqrt{\frac{787}{16000}+0.4\times \frac{1}{1600}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Multiply 0.4 and \frac{1}{1600} to get \frac{1}{4000}.

$$\sqrt{\frac{787}{16000}+\frac{1}{4000}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Add \frac{787}{16000} and \frac{1}{4000} to get \frac{791}{16000}.

$$\sqrt{\frac{791}{16000}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Expand \frac{5.5}{100} by multiplying both numerator and the denominator by 10.

$$\sqrt{\frac{791}{16000}+0.2\times \left(\frac{55}{1000}\right)^{2}}$$

Reduce the fraction \frac{55}{1000} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{791}{16000}+0.2\times \left(\frac{11}{200}\right)^{2}}$$

Calculate \frac{11}{200} to the power of 2 and get \frac{121}{40000}.

$$\sqrt{\frac{791}{16000}+0.2\times \frac{121}{40000}}$$

Multiply 0.2 and \frac{121}{40000} to get \frac{121}{200000}.

$$\sqrt{\frac{791}{16000}+\frac{121}{200000}}$$

Add \frac{791}{16000} and \frac{121}{200000} to get \frac{20017}{400000}.

$$\sqrt{\frac{20017}{400000}}$$

Rewrite the square root of the division \sqrt{\frac{20017}{400000}} as the division of square roots \frac{\sqrt{20017}}{\sqrt{400000}}.

$$\frac{\sqrt{20017}}{\sqrt{400000}}$$

Factor 400000=200^{2}\times 10. Rewrite the square root of the product \sqrt{200^{2}\times 10} as the product of square roots \sqrt{200^{2}}\sqrt{10}. Take the square root of 200^{2}.

$$\frac{\sqrt{20017}}{200\sqrt{10}}$$

Rationalize the denominator of \frac{\sqrt{20017}}{200\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\sqrt{20017}\sqrt{10}}{200\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\sqrt{20017}\sqrt{10}}{200\times 10}$$

To multiply \sqrt{20017} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\sqrt{200170}}{200\times 10}$$

Multiply 200 and 10 to get 2000.

$$\frac{\sqrt{200170}}{2000}$$