Solve sqrt{0.1(-14.5%)^2+0.3(-2.5%)^2+0.4(2.5%)^2+0.2(5.5%)^2}

$$\sqrt { 0.1 ( - 14.5 \% ) ^ { 2 } + 0.3 ( - 2.5 \% ) ^ { 2 } + 0.4 ( 2.5 \% ) ^ { 2 } + 0.2 ( 5.5 \% ) ^ { 2 } }$$

Evaluate

$$\frac{\sqrt{3145}}{1000}\approx 0.0560803$$

Solution Steps

Expand \frac{14.5}{100} by multiplying both numerator and the denominator by 10.

$$\sqrt{0.1\left(-\frac{145}{1000}\right)^{2}+0.3\left(-\frac{2.5}{100}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Reduce the fraction \frac{145}{1000} to lowest terms by extracting and canceling out 5.

$$\sqrt{0.1\left(-\frac{29}{200}\right)^{2}+0.3\left(-\frac{2.5}{100}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Calculate -\frac{29}{200} to the power of 2 and get \frac{841}{40000}.

$$\sqrt{0.1\times \frac{841}{40000}+0.3\left(-\frac{2.5}{100}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Multiply 0.1 and \frac{841}{40000} to get \frac{841}{400000}.

$$\sqrt{\frac{841}{400000}+0.3\left(-\frac{2.5}{100}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{841}{400000}+0.3\left(-\frac{25}{1000}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{841}{400000}+0.3\left(-\frac{1}{40}\right)^{2}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{841}{400000}+0.3\times \frac{1}{1600}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{841}{400000}+\frac{3}{16000}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Add \frac{841}{400000} and \frac{3}{16000} to get \frac{229}{100000}.

$$\sqrt{\frac{229}{100000}+0.4\times \left(\frac{2.5}{100}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{229}{100000}+0.4\times \left(\frac{25}{1000}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{229}{100000}+0.4\times \left(\frac{1}{40}\right)^{2}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{229}{100000}+0.4\times \frac{1}{1600}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{229}{100000}+\frac{1}{4000}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

Add \frac{229}{100000} and \frac{1}{4000} to get \frac{127}{50000}.

$$\sqrt{\frac{127}{50000}+0.2\times \left(\frac{5.5}{100}\right)^{2}}$$

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

$$\sqrt{\frac{127}{50000}+0.2\times \left(\frac{55}{1000}\right)^{2}}$$

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

$$\sqrt{\frac{127}{50000}+0.2\times \left(\frac{11}{200}\right)^{2}}$$

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

$$\sqrt{\frac{127}{50000}+0.2\times \frac{121}{40000}}$$

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

$$\sqrt{\frac{127}{50000}+\frac{121}{200000}}$$

Add \frac{127}{50000} and \frac{121}{200000} to get \frac{629}{200000}.

$$\sqrt{\frac{629}{200000}}$$

Rewrite the square root of the division \sqrt{\frac{629}{200000}} as the division of square roots \frac{\sqrt{629}}{\sqrt{200000}}.

$$\frac{\sqrt{629}}{\sqrt{200000}}$$

Factor 200000=200^{2}\times 5. Rewrite the square root of the product \sqrt{200^{2}\times 5} as the product of square roots \sqrt{200^{2}}\sqrt{5}. Take the square root of 200^{2}.

$$\frac{\sqrt{629}}{200\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{629}}{200\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{629}\sqrt{5}}{200\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{629}\sqrt{5}}{200\times 5}$$

To multiply \sqrt{629} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{3145}}{200\times 5}$$

Multiply 200 and 5 to get 1000.

$$\frac{\sqrt{3145}}{1000}$$