Question

express each number in decimal notation.

1. $2.3\times10^{5}$
2. $6.5\times10^{4}$
3. $3\times10^{3}$
4. $4\times10^{5}$
5. $6.05\times10^{-1}$
6. $4.15\times10^{-3}$
7. $3\times10^{-3}$
8. $4\times10^{-5}$
9. $7.5\times10^{0}$
10. $8.2\times10^{0}$

express the number in scientific notation.

1. 650
2. 724
3. 3,200,000,000
4. 14,800,000
5. 0.05
6. 0.0025
7. 0.00000075
8. 0.00001203
9. 6.5
10. 8
11. distance from the sun to the earth: 147,000,000 km
12. mass of an oxygen molecule: 0.00000000000000000000053 g

multiply and write the answers in scientific notation.

1. $(0.00000008)(6000)$
2. $(750,000)(23,000,000)$
3. $(2.45\times10^{-2})(5.1\times10^{-8})$
4. $(4.05\times10^{-6})(8.001\times10^{8})$
5. $(3.5\times10^{0})(1.25\times10^{-10})$
6. $(1.415\times10^{-2})(6.1\times10^{0})$
7. $(5.15\times10^{12})(9.1\times10^{8})$
8. $(6.25\times10^{2})(3.215\times10^{9})$
9. approximately how many atoms are in one gram of pure gold? (see example 4 for help with this problem.)
10. approximately how much would $8.0\times10^{24}$ atoms of silver weigh, in grams? (see example 4 for help with this problem.)

Explanation:

Step1: Recall scientific - notation rules

Scientific notation is of the form $a\times10^{n}$, where $1\leq|a|\lt10$ and $n$ is an integer. To convert from scientific notation to decimal notation: if $n\gt0$, move the decimal point $n$ places to the right; if $n\lt0$, move the decimal point $|n|$ places to the left. To multiply numbers in scientific notation, multiply the $a$ - values and add the exponents of 10.

Step2: Convert to decimal notation (1 - 10)

1. For $2.3\times10^{5}$, since $n = 5\gt0$, move the decimal point 5 places to the right: $2.3\times10^{5}=230000$.
2. For $6.5\times10^{4}$, since $n = 4\gt0$, move the decimal point 4 places to the right: $6.5\times10^{4}=65000$.
3. For $3\times10^{3}$, since $n = 3\gt0$, move the decimal point 3 places to the right: $3\times10^{3}=3000$.
4. For $4\times10^{5}$, since $n = 5\gt0$, move the decimal point 5 places to the right: $4\times10^{5}=400000$.
5. For $6.05\times10^{-1}$, since $n=-1\lt0$, move the decimal point 1 place to the left: $6.05\times10^{-1}=0.605$.
6. For $4.15\times10^{-3}$, since $n = - 3\lt0$, move the decimal point 3 places to the left: $4.15\times10^{-3}=0.00415$.
7. For $3\times10^{-3}$, since $n=-3\lt0$, move the decimal point 3 places to the left: $3\times10^{-3}=0.003$.
8. For $4\times10^{-5}$, since $n=-5\lt0$, move the decimal point 5 places to the left: $4\times10^{-5}=0.00004$.
9. For $7.5\times10^{0}$, since $10^{0}=1$, then $7.5\times10^{0}=7.5$.
10. For $8.2\times10^{0}$, since $10^{0}=1$, then $8.2\times10^{0}=8.2$.

Step3: Convert to scientific notation (11 - 17)

1. For 650, move the decimal point 2 places to the left: $650 = 6.5\times10^{2}$.
2. For 724, move the decimal point 2 places to the left: $724=7.24\times10^{2}$.
3. For 3200000000, move the decimal point 9 places to the left: $3200000000 = 3.2\times10^{9}$.
4. For 14800000, move the decimal point 7 places to the left: $14800000=1.48\times10^{7}$.
5. For 0.05, move the decimal point 2 places to the right: $0.05 = 5\times10^{-2}$.
6. For 0.0025, move the decimal point 3 places to the right: $0.0025=2.5\times10^{-3}$.
7. For 0.00000075, move the decimal point 7 places to the right: $0.00000075 = 7.5\times10^{-7}$.

Step4: Multiply and write in scientific notation (23 - 30)

1. First, rewrite $0.00000008 = 8\times10^{-8}$ and $6000 = 6\times10^{3}$. Then $(8\times10^{-8})(6\times10^{3})=(8\times6)\times10^{-8 + 3}=48\times10^{-5}=4.8\times10^{-4}$.
2. Rewrite $750000 = 7.5\times10^{5}$ and $23000000=2.3\times10^{7}$. Then $(7.5\times10^{5})(2.3\times10^{7})=(7.5\times2.3)\times10^{5 + 7}=17.25\times10^{12}=1.725\times10^{13}$.
3. $(2.45\times10^{-2})(5.1\times10^{-8})=(2.45\times5.1)\times10^{-2-8}=12.495\times10^{-10}=1.2495\times10^{-9}$.
4. $(4.05\times10^{-6})(8.001\times10^{8})=(4.05\times8.001)\times10^{-6 + 8}=32.40405\times10^{2}=3.240405\times10^{3}$.
5. $(3.5\times10^{0})(1.25\times10^{-10})=(3.5\times1.25)\times10^{0-10}=4.375\times10^{-10}$.
6. $(1.415\times10^{-2})(6.1\times10^{0})=(1.415\times6.1)\times10^{-2+0}=8.6315\times10^{-2}$.
7. $(5.15\times10^{12})(9.1\times10^{8})=(5.15\times9.1)\times10^{12 + 8}=46.865\times10^{20}=4.6865\times10^{21}$.
8. $(6.25\times10^{2})(3.215\times10^{9})=(6.25\times3.215)\times10^{2 + 9}=20.09375\times10^{11}=2.009375\times10^{12}$.

Answer:

1. $230000$
2. $65000$
3. $3000$
4. $400000$
5. $0.605$
6. $0.00415$
7. $0.003$
8. $0.00004$
9. $7.5$
10. $8.2$
11. $6.5\times10^{2}$
12. $7.24\times10^{2}$
13. $3.2\times10^{9}$
14. $1.48\times10^{7}$
15. $5\times10^{-2}$
16. $2.5\times10^{-3}$
17. $7.5\times10^{-7}$
18. $4.8\times10^{-4}$
19. $1.725\times10^{13}$
20. $1.2495\times10^{-9}$
21. $3.240405\times10^{3}$
22. $4.375\times10^{-10}$
23. $8.6315\times10^{-2}$
24. $4.6865\times10^{21}$
25. $2.009375\times10^{12}$