3.7 Scientific notation  (Page 2/3)

$0.000000000004632=4.632\times {10}^{-12}$

The decimal point is twelve places to the right of its original position, and the power of 10 is $-12$ .

$0.027=2.7\times {10}^{-2}$

The decimal point is two places to the right of its original position, and the power of 10 is $-2$ .

$4.673\times {10}^4$ .

The exponent of 10 is 4 so we must move the decimal point to the right 4 places (adding $0\text{'}\text{s}$ if necessary).

$2.9\times {10}^7$ .

The exponent of 10 is 7 so we must move the decimal point to the right 7 places (adding $0\text{'}\text{s}$ if necessary).

$2.9\times {10}^7=\text{29000000}$

$1\times {10}^{27}$ .

The exponent of 10 is 27 so we must move the decimal point to the right 27 places (adding $0\text{'}\text{s}$ without a doubt).

$1\times {10}^{27}=\text{1,000,000,000,000,000,000,000,000,000}$

$4.21\times {10}^{-5}$ .

The exponent of 10 is $-5$ so we must move the decimal point to the left 5 places (adding $0\text{'}\text{s}$ if necessary).

$4.21\times {10}^{-5}=\text{0}\text{.0000421}$

$1.006\times {10}^{-18}$ .

The exponent of 10 is $-18$ so we must move the decimal point to the left 18 places (adding $0\text{'}\text{s}$ if necessary).

$1.006\times {10}^{-18}=0.000000000000000001006$

$\begin{array}{ll}(2\times {10}^3)(4\times {10}^8)\hfill & =(2\times 4)({10}^3\times {10}^8)\hfill \\ \hfill & =8\times {10}^{3+8}\hfill \\ \hfill & =8\times {10}^{11}\hfill \end{array}$

$\begin{array}{ll}(5\times {10}^{17})(8.1\times {10}^{-22})\hfill & =(5\times 8.1)({10}^{17}\times {10}^{-22})\hfill \\ \hfill & =40.5\times {10}^{17-22}\hfill \\ \hfill & =40.5\times {10}^{-5}\hfill \end{array}$

We need to move the decimal point one place to the left to put this number in scientific notation.

Thus, we must also change the exponent of 10.

$\begin{array}{l}40.5\times {10}^{-5}\hfill \\ 4.05\times {10}^1\times {10}^{-5}\hfill \\ 4.05\times ({10}^1\times {10}^{-5})\hfill \\ 4.05\times ({10}^{1-5})\hfill \\ 4.05\times {10}^{-4}\hfill \end{array}$

Thus,

$(5\times {10}^{17})(8.1\times {10}^{-22})=4.05\times {10}^{-4}$