Question

scientific measurements and calculations unit test
express the following numbers using scientific notation:

1. 7,830
2. 0.000742
3. 2,005,000
4. 0.00307
5. 0.068
6. 0.00000096

express the following as whole numbers or decimals:

1. 4.82×10³
2. 9.07×10⁻³
3. 6.02×10⁵
4. 3.5×10⁻⁴
5. 8.20×10²
6. 7.006×10⁰

perform the following calculations and express the answers in scientific notation.

1. (4.2×10³)+(6.8×10²)
2. (0.000 096 0)÷(0.000 002 4)
3. (-5.8×10⁴)×(4.1×10⁻³)
4. (9.3×10²)-(7.25×10¹)
5. (0.000 003 42)(2.6×10⁶)(0.987)
6. (0.000 452 1)/(5.63×10⁻⁵)(3.28×10³)

Explanation:

Step1: Convert to scientific - notation (general form $a\times10^{n}$, $1\leqslant|a|\lt10$)

1. For $7830$, move the decimal 3 places to the left: $7.83\times 10^{3}$
2. For $0.000742$, move the decimal 4 places to the right: $7.42\times 10^{-4}$
3. For $2005000$, move the decimal 6 places to the left: $2.005\times 10^{6}$
4. For $0.00307$, move the decimal 3 places to the right: $3.07\times 10^{-3}$
5. For $0.068$, move the decimal 2 places to the right: $6.8\times 10^{-2}$
6. For $0.00000096$, move the decimal 7 places to the right: $9.6\times 10^{-7}$

Step2: Convert from scientific - notation to decimal

1. For $4.82\times 10^{3}$, move the decimal 3 places to the right: $4820$
2. For $9.07\times 10^{-3}$, move the decimal 3 places to the left: $0.00907$
3. For $6.02\times 10^{5}$, move the decimal 5 places to the right: $602000$
4. For $3.5\times 10^{-4}$, move the decimal 4 places to the left: $0.00035$
5. For $8.20\times 10^{2}$, move the decimal 2 places to the right: $820$
6. For $7.006\times 10^{0}$, since $10^{0}=1$, the number is $7.006$

Step3: Perform calculations in scientific - notation

1. $(4.2\times 10^{3})+(6.8\times 10^{2})=(4.2\times 10^{3})+(0.68\times 10^{3})=(4.2 + 0.68)\times10^{3}=4.88\times 10^{3}$
2. $(0.0000960)\div(0.0000024)=\frac{9.6\times 10^{-5}}{2.4\times 10^{-6}}=\frac{9.6}{2.4}\times10^{-5+6}=4\times 10^{1}$
3. $(-5.8\times 10^{4})\times(4.1\times 10^{-3})=(-5.8\times4.1)\times10^{4 - 3}=-23.78\times 10^{1}=-2.378\times 10^{2}$
4. $(9.3\times 10^{2})-(7.25\times 10^{1})=(9.3\times 10^{2})-(0.725\times 10^{2})=(9.3 - 0.725)\times10^{2}=8.575\times 10^{2}$
5. $(0.00000342)\times(2.6\times 10^{6})\times(0.987)=(3.42\times 10^{-6})\times(2.6\times 10^{6})\times(0.987)=(3.42\times2.6\times0.987)\times10^{-6 + 6}=8.776404\times 10^{0}$
6. $(0.0004521)\div[(5.63\times 10^{-5})\times(3.28\times 10^{3})]=\frac{4.521\times 10^{-4}}{(5.63\times3.28)\times10^{-5 + 3}}=\frac{4.521\times 10^{-4}}{18.4664\times 10^{-2}}=\frac{4.521}{18.4664}\times10^{-4+2}\approx0.245\times 10^{-2}=2.45\times 10^{-3}$

Answer:

1. $7.83\times 10^{3}$
2. $7.42\times 10^{-4}$
3. $2.005\times 10^{6}$
4. $3.07\times 10^{-3}$
5. $6.8\times 10^{-2}$
6. $9.6\times 10^{-7}$
7. $4820$
8. $0.00907$
9. $602000$
10. $0.00035$
11. $820$
12. $7.006$
13. $4.88\times 10^{3}$
14. $4\times 10^{1}$
15. $-2.378\times 10^{2}$
16. $8.575\times 10^{2}$
17. $8.776404\times 10^{0}$
18. $2.45\times 10^{-3}$