Solve {l}{(4*10^{-3})(2*10^{5})}{(3*10^{-1})(6*10^{-8})}{(6.7*10^{3})(4.0*10^5)}{(3.7*10^-5)(200)}{(4*10^-3)^2}

$$\left. \begin{array} { l } { ( 4 \times 10 ^ { - 3 } ) ( 2 \times 10 ^ { 5 } ) } \\ { ( 3 \times 10 ^ { - 1 } ) ( 6 \times 10 ^ { - 8 } ) } \\ { ( 6.7 \times 10 ^ { 3 } ) ( 4.0 \times 10 ^ { 5 } ) } \\ { ( 3.7 \times 10 ^ { - 5 } ) ( 200 ) } \\ { ( 4 \times 10 ^ { - 3 } ) ^ { 2 } } \end{array} \right.$$

Sort

$$0.000000018,0.000016,0.0074,800,2680000000$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -3 and 5 to get 2.

$$sort(4\times 10^{2}\times 2,3\times 10^{-1}\times 6\times 10^{-8},6.7\times 10^{3}\times 4\times 10^{5},3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

To multiply powers of the same base, add their exponents. Add -1 and -8 to get -9.

$$sort(4\times 10^{2}\times 2,3\times 10^{-9}\times 6,6.7\times 10^{3}\times 4\times 10^{5},3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$sort(4\times 10^{2}\times 2,3\times 10^{-9}\times 6,6.7\times 10^{8}\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Calculate 10 to the power of 2 and get 100.

$$sort(4\times 100\times 2,3\times 10^{-9}\times 6,6.7\times 10^{8}\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Multiply 4 and 100 to get 400.

$$sort(400\times 2,3\times 10^{-9}\times 6,6.7\times 10^{8}\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Multiply 400 and 2 to get 800.

$$sort(800,3\times 10^{-9}\times 6,6.7\times 10^{8}\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Calculate 10 to the power of -9 and get \frac{1}{1000000000}.

$$sort(800,3\times \frac{1}{1000000000}\times 6,6.7\times 10^{8}\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Multiply 3 and \frac{1}{1000000000} to get \frac{3}{1000000000}.

$$sort(800,\frac{3}{1000000000}\times 6,6.7\times 10^{8}\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Multiply \frac{3}{1000000000} and 6 to get \frac{9}{500000000}.

$$sort(800,\frac{9}{500000000},6.7\times 10^{8}\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Calculate 10 to the power of 8 and get 100000000.

$$sort(800,\frac{9}{500000000},6.7\times 100000000\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Multiply 6.7 and 100000000 to get 670000000.

$$sort(800,\frac{9}{500000000},670000000\times 4,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Multiply 670000000 and 4 to get 2680000000.

$$sort(800,\frac{9}{500000000},2680000000,3.7\times 10^{-5}\times 200,\left(4\times 10^{-3}\right)^{2})$$

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

$$sort(800,\frac{9}{500000000},2680000000,3.7\times \frac{1}{100000}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Multiply 3.7 and \frac{1}{100000} to get \frac{37}{1000000}.

$$sort(800,\frac{9}{500000000},2680000000,\frac{37}{1000000}\times 200,\left(4\times 10^{-3}\right)^{2})$$

Multiply \frac{37}{1000000} and 200 to get \frac{37}{5000}.

$$sort(800,\frac{9}{500000000},2680000000,\frac{37}{5000},\left(4\times 10^{-3}\right)^{2})$$

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

$$sort(800,\frac{9}{500000000},2680000000,\frac{37}{5000},\left(4\times \frac{1}{1000}\right)^{2})$$

Multiply 4 and \frac{1}{1000} to get \frac{1}{250}.

$$sort(800,\frac{9}{500000000},2680000000,\frac{37}{5000},\left(\frac{1}{250}\right)^{2})$$

Calculate \frac{1}{250} to the power of 2 and get \frac{1}{62500}.

$$sort(800,\frac{9}{500000000},2680000000,\frac{37}{5000},\frac{1}{62500})$$

Convert decimal numbers in the list 800,\frac{9}{500000000},2680000000,\frac{37}{5000},\frac{1}{62500} to fractions.

$$800,\frac{9}{500000000},2680000000,\frac{37}{5000},\frac{1}{62500}$$

Least common denominator of the numbers in the list 800,\frac{9}{500000000},2680000000,\frac{37}{5000},\frac{1}{62500} is 500000000. Convert numbers in the list to fractions with denominator 500000000.

$$\frac{400000000000}{500000000},\frac{9}{500000000},\frac{1340000000000000000}{500000000},\frac{3700000}{500000000},\frac{8000}{500000000}$$

To sort the list, start from a single element \frac{400000000000}{500000000}.

$$\frac{400000000000}{500000000}$$

Insert \frac{9}{500000000} to the appropriate location in the new list.

$$\frac{9}{500000000},\frac{400000000000}{500000000}$$

Insert \frac{1340000000000000000}{500000000} to the appropriate location in the new list.

$$\frac{9}{500000000},\frac{400000000000}{500000000},\frac{1340000000000000000}{500000000}$$

Insert \frac{3700000}{500000000} to the appropriate location in the new list.

$$\frac{9}{500000000},\frac{3700000}{500000000},\frac{400000000000}{500000000},\frac{1340000000000000000}{500000000}$$

Insert \frac{8000}{500000000} to the appropriate location in the new list.

$$\frac{9}{500000000},\frac{8000}{500000000},\frac{3700000}{500000000},\frac{400000000000}{500000000},\frac{1340000000000000000}{500000000}$$

Replace the obtained fractions with the initial values.

$$\frac{9}{500000000},\frac{1}{62500},\frac{37}{5000},800,2680000000$$

Evaluate

$$800,\ 0.000000018,\ 2680000000,\ 0.0074,\ 0.000016$$