Solve sqrt{0.1(-31%)^2+0.3(-11%)^2+0.4(4%)^2+0.2(24%)^2}

$$\sqrt { 0.1 ( - 31 \% ) ^ { 2 } + 0.3 ( - 11 \% ) ^ { 2 } + 0.4 ( 4 \% ) ^ { 2 } + 0.2 ( 24 \% ) ^ { 2 } }$$

Evaluate

$$\frac{\sqrt{254}}{100}\approx 0.159373775$$

Solution Steps

Calculate -\frac{31}{100} to the power of 2 and get \frac{961}{10000}.

$$\sqrt{0.1\times \frac{961}{10000}+0.3\left(-\frac{11}{100}\right)^{2}+0.4\times \left(\frac{4}{100}\right)^{2}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Multiply 0.1 and \frac{961}{10000} to get \frac{961}{100000}.

$$\sqrt{\frac{961}{100000}+0.3\left(-\frac{11}{100}\right)^{2}+0.4\times \left(\frac{4}{100}\right)^{2}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Calculate -\frac{11}{100} to the power of 2 and get \frac{121}{10000}.

$$\sqrt{\frac{961}{100000}+0.3\times \frac{121}{10000}+0.4\times \left(\frac{4}{100}\right)^{2}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Multiply 0.3 and \frac{121}{10000} to get \frac{363}{100000}.

$$\sqrt{\frac{961}{100000}+\frac{363}{100000}+0.4\times \left(\frac{4}{100}\right)^{2}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Add \frac{961}{100000} and \frac{363}{100000} to get \frac{331}{25000}.

$$\sqrt{\frac{331}{25000}+0.4\times \left(\frac{4}{100}\right)^{2}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Reduce the fraction \frac{4}{100} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{331}{25000}+0.4\times \left(\frac{1}{25}\right)^{2}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Calculate \frac{1}{25} to the power of 2 and get \frac{1}{625}.

$$\sqrt{\frac{331}{25000}+0.4\times \frac{1}{625}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Multiply 0.4 and \frac{1}{625} to get \frac{2}{3125}.

$$\sqrt{\frac{331}{25000}+\frac{2}{3125}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Add \frac{331}{25000} and \frac{2}{3125} to get \frac{347}{25000}.

$$\sqrt{\frac{347}{25000}+0.2\times \left(\frac{24}{100}\right)^{2}}$$

Reduce the fraction \frac{24}{100} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{347}{25000}+0.2\times \left(\frac{6}{25}\right)^{2}}$$

Calculate \frac{6}{25} to the power of 2 and get \frac{36}{625}.

$$\sqrt{\frac{347}{25000}+0.2\times \frac{36}{625}}$$

Multiply 0.2 and \frac{36}{625} to get \frac{36}{3125}.

$$\sqrt{\frac{347}{25000}+\frac{36}{3125}}$$

Add \frac{347}{25000} and \frac{36}{3125} to get \frac{127}{5000}.

$$\sqrt{\frac{127}{5000}}$$

Rewrite the square root of the division \sqrt{\frac{127}{5000}} as the division of square roots \frac{\sqrt{127}}{\sqrt{5000}}.

$$\frac{\sqrt{127}}{\sqrt{5000}}$$

Factor 5000=50^{2}\times 2. Rewrite the square root of the product \sqrt{50^{2}\times 2} as the product of square roots \sqrt{50^{2}}\sqrt{2}. Take the square root of 50^{2}.

$$\frac{\sqrt{127}}{50\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{127}}{50\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{127}\sqrt{2}}{50\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{127}\sqrt{2}}{50\times 2}$$

To multiply \sqrt{127} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{254}}{50\times 2}$$

Multiply 50 and 2 to get 100.

$$\frac{\sqrt{254}}{100}$$