1. Write each of the following numbers in the form k x 10n where 1≤ k < 10 and n is an integer.
a. 1,384,000
\begin{align*} 1,384,000 \end{align*}
Solution:
\begin{align*} & = 1.384000 \times 10^6 \end{align*}
b. 12.32005
\begin{align*} 12.32005 \end{align*}
Solution:
\begin{align*} & = 1.232005 \times 10^1 \end{align*}
c. 2157.957
\begin{align*} 2157.957 \end{align*}
Solution:
\begin{align*} & = 2.157957 \times 10^3 \end{align*}
d. 0.00002
\begin{align*} 0.00002 \end{align*}
Solution:
\begin{align*} & = 2 \times 10^{-5} \end{align*}
e. 0.00729
\begin{align*} 0.00729 \end{align*}
Solution:
\begin{align*} & = 7.29 \times 10^{-3} \end{align*}
f. 0.000000000926
\begin{align*} 0.000000000926 \end{align*}
Solution:
\begin{align*} & = 9.26 \times 10^{-10} \end{align*}
g. 520,000,000
\begin{align*} 520,000,000 \end{align*}
Solution:
\begin{align*} & = 5.2 \times 10^{8} \end{align*}
h. 0.0000085
\begin{align*} 0.0000085 \end{align*}
Solution:
\begin{align*} & = 8.5 \times 10^{-6} \end{align*}
2. Write the following numbers in the usual form.
a. 52.5 x 104
\begin{align*} 52.5 \times 10^4 \end{align*}
Solution:
\begin{align*} &= 52.5 \times 10000 \\ &= 525000 \end{align*}
b. 158.9 x 106
\begin{align*} 158.9 \times 10^6 \end{align*}
Solution:
\begin{align*} &= 158.9 \times 1000000 \\ &= 158900000 \end{align*}
c. 9.545 x 1012
\begin{align*} 9.545 \times 10^{12} \end{align*}
Solution:
\begin{align*} &= 9.545 \times 1000000000000 \\ &= 9545000000000 \end{align*}
d. 1.72 x 10-5
\begin{align*} 1.72 \times 10^{-5} \end{align*}
Solution:
\begin{align*} &= 1.72 \times \frac{1}{10^{5}} \\ \\ &= 1.72 \times \frac{1}{100000} \\ \\ &= 0.0000172 \end{align*}
e. 8.5 x 10-7
\begin{align*} 8.5 \times 10^{-7} \end{align*}
Solution:
\begin{align*} &= 8.5 \times \frac{1}{10^{7}} \\ \\ &= 8.5 \times \frac{1}{10000000} \\ \\ &= 0.00000085 \end{align*}
f. 2.9 x 10-9
\begin{align*} 2.9 \times 10^{-9} \end{align*}
Solution:
\begin{align*} &= 2.9 \times \frac{1}{10^{9}} \\ \\ &= 2.9 \times \frac{1}{1000000000} \\ \\ &= 0.0000000029 \end{align*}
g. 6293.2 x 105
\begin{align*} 6293.2 \times 10^{5} \end{align*}
Solution:
\begin{align*} &= 6293.2 \times 10^5 \\ \\ &= 6293.2 \times 100000 \\ \\ &= 629320000 \end{align*}
h. 1.925 x 10-6
\begin{align*} 1.925 \times 10^{-6} \end{align*}
Solution:
\begin{align*} &= 1.925 \times \frac{1}{10^6} \\ \\ &= 1.925 \times \frac{1}{100000} \\ \\ &= 0.00001925 \end{align*}
3. Express each of the following in the form k x 10n; (1 ≤ k < 10)
a. \((1.25 \times 10^7) \div (5 \times 10^4)\)
Solution:
\begin{align*} &= \frac{1.25}{5} \times \frac{10^7}{10^3} \\ \\ &= 0.25 \times 10^{7-3} \\ \\ &= 0.25 \times 10^{4} \\ \\ &= 2.5 \times 10^3 \\ \\ \end{align*}
b. \((2.5 \times 10^{10}) \times (31.25 \times 10^{-5})\)
Solution:
\begin{align*} &= 2.5 \times 10^{10} \times 31.25 \times 10^{-5}\\ &= 2.5 \times 31.25 \times 10^{10} \times 10^{-5}\\ &= 78.125 \times 10^{10 + (-5)}\\ &= 78.125 \times 10^{10 - 5}\\ &= 78.125 \times 10^{5}\\ &= 7.8125 \times 10^{6}\\ \end{align*}
c. \((1.6 \times 10^{9}) \times (5.0 \times 10^{-3})\)
Solution:
\begin{align*} &= 1.6 \times 10^{9} \times 5.0 \times 10^{-3}\\ &= 1.6 \times 5.0 \times 10^{9} \times 10^{-3}\\ &= 8.00 \times 10^{9 + (-3)}\\ &= 8.0 \times 10^{9 - 3}\\ &= 8.0 \times 10^{6}\\ \end{align*}
d. \((3.4 \times 10^{4}) \times (5 \times 10^{-3}) \div [2.0 \times 10^5]\)
Solution:
\begin{align*} &= \frac{(3.4 \times 10^{4}) \times (5 \times 10^{-3})}{2.0 \times 10^5} \\ \\ &= \frac{3.4 \times 10^{4} \times 5 \times 10^{-3}}{2 \times 10^5} \\ \\ &= \frac{\cancelto{1.7}{3.4} \times 5 \times 10^{4} \times 10^{-3}}{\cancel2 \times 10^5} \\ \\ &= \frac{{1.7} \times 5 \times 10^{4} \times 10^{-3}}{10^5} \\ \\ &= 8.5 \times 10^{4 +(-3) -5} \\ &= 8.5 \times 10^{4 -3 -5} \\ &= 8.5 \times 10^{1 -5} \\ &= 8.5 \times 10^{-4} \\ \end{align*}
4. Express the numbers appearing in the following statements in the form k x 10n where 1 ≤ k < 10 and n is an integer.
a. Sun's diameter is \(1,384,000 \) km.
Solution:
\begin{align*} &= 1,384,000 \\ &= 1.384000 \times 10^6 \\ &= 1.384 \times 10^6 \\ \end{align*}
b. The distance of the sun from the earth is approximately \(150,000,000 \) km.
Solution:
\begin{align*} &= 150,000,000 \\ &= 1.50000000 \times 10^8 \\ &= 1.5 \times 10^8 \\ \end{align*}
c. The speed of the light is about \(27600 \, \text{km/sec.}\)
Solution:
\begin{align*} &= 27600 \\ &= 2.7600 \times 10^4 \\ &= 2.76 \times 10^4 \\ \end{align*}
d. 1 Angstrom unit = \(\frac{1}{10,000,000,000} \, m\).
Solution:
\begin{align*} &= \frac{1}{10,000,000,000} \\ \\ &= \frac{1}{10^{10}} \\ \\ &= 1 \times 10^{-10} \\ &= 1.0 \times 10^{-10} \\ \end{align*}