Solve sqrt{[(5.32*4.7(10^{-3}))^{2}+(-1.7*10^{-3}*7.7*10^-4)^2+(-8732.2*2*10^-5)^2]}

$$\sqrt { [ ( 5.32 * 4.7 ( 10 ^ { - 3 } ) ) ^ { 2 } + ( - 1.7 \times 10 ^ { - 3 } * 7.7 \times 10 ^ { - 4 } ) ^ { 2 } + ( - 8732.2 * 2 \times 10 ^ { - 5 } ) ^ { 2 } ] }$$

Evaluate

$$\frac{\sqrt{31125726753713481}}{1000000000}\approx 0.176424847$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -3 and -4 to get -7.

$$\sqrt{\left(5.32\times 4.7\times 10^{-3}\right)^{2}+\left(-1.7\times 10^{-7}\times 7.7\right)^{2}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Multiply 5.32 and 4.7 to get 25.004.

$$\sqrt{\left(25.004\times 10^{-3}\right)^{2}+\left(-1.7\times 10^{-7}\times 7.7\right)^{2}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$\sqrt{\left(25.004\times \frac{1}{1000}\right)^{2}+\left(-1.7\times 10^{-7}\times 7.7\right)^{2}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Multiply 25.004 and \frac{1}{1000} to get \frac{6251}{250000}.

$$\sqrt{\left(\frac{6251}{250000}\right)^{2}+\left(-1.7\times 10^{-7}\times 7.7\right)^{2}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Calculate \frac{6251}{250000} to the power of 2 and get \frac{39075001}{62500000000}.

$$\sqrt{\frac{39075001}{62500000000}+\left(-1.7\times 10^{-7}\times 7.7\right)^{2}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$\sqrt{\frac{39075001}{62500000000}+\left(-1.7\times \frac{1}{10000000}\times 7.7\right)^{2}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Multiply -1.7 and \frac{1}{10000000} to get -\frac{17}{100000000}.

$$\sqrt{\frac{39075001}{62500000000}+\left(-\frac{17}{100000000}\times 7.7\right)^{2}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Multiply -\frac{17}{100000000} and 7.7 to get -\frac{1309}{1000000000}.

$$\sqrt{\frac{39075001}{62500000000}+\left(-\frac{1309}{1000000000}\right)^{2}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Calculate -\frac{1309}{1000000000} to the power of 2 and get \frac{1713481}{1000000000000000000}.

$$\sqrt{\frac{39075001}{62500000000}+\frac{1713481}{1000000000000000000}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Add \frac{39075001}{62500000000} and \frac{1713481}{1000000000000000000} to get \frac{625200017713481}{1000000000000000000}.

$$\sqrt{\frac{625200017713481}{1000000000000000000}+\left(-8732.2\times 2\times 10^{-5}\right)^{2}}$$

Multiply -8732.2 and 2 to get -17464.4.

$$\sqrt{\frac{625200017713481}{1000000000000000000}+\left(-17464.4\times 10^{-5}\right)^{2}}$$

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$\sqrt{\frac{625200017713481}{1000000000000000000}+\left(-17464.4\times \frac{1}{100000}\right)^{2}}$$

Multiply -17464.4 and \frac{1}{100000} to get -\frac{43661}{250000}.

$$\sqrt{\frac{625200017713481}{1000000000000000000}+\left(-\frac{43661}{250000}\right)^{2}}$$

Calculate -\frac{43661}{250000} to the power of 2 and get \frac{1906282921}{62500000000}.

$$\sqrt{\frac{625200017713481}{1000000000000000000}+\frac{1906282921}{62500000000}}$$

Add \frac{625200017713481}{1000000000000000000} and \frac{1906282921}{62500000000} to get \frac{31125726753713481}{1000000000000000000}.

$$\sqrt{\frac{31125726753713481}{1000000000000000000}}$$

Rewrite the square root of the division \sqrt{\frac{31125726753713481}{1000000000000000000}} as the division of square roots \frac{\sqrt{31125726753713481}}{\sqrt{1000000000000000000}}.

$$\frac{\sqrt{31125726753713481}}{\sqrt{1000000000000000000}}$$

Calculate the square root of 1000000000000000000 and get 1000000000.

$$\frac{\sqrt{31125726753713481}}{1000000000}$$