Perform Operations With Scientific Notation - 8th Grade Math

1. What is $(6\times 10^2)(0.004)$ in scientific notation?

   $2.4\times 10^0$. $(6 \times 10^2)(4 \times 10^{-3}) = 24 \times 10^{-1} = 2.4 \times 10^0$.

2. What is the number $12\times 10^3$ written in proper scientific notation?

   $1.2\times 10^4$. Move decimal left: $12 = 1.2×10^1$, so $12×10^3 = 1.2×10^4$.

3. Choose the more appropriate unit for seafloor spreading of $0.03\ \text{m/year}$: $\text{m/year}$ or $\text{mm/year}$?

   $\text{mm/year}$. Converting to mm gives 30 mm/year, avoiding small decimals.

4. What does a calculator display of $1.23\text{E}5$ mean in scientific notation?

   $1.23\times 10^5$. E notation means "times 10 to the power of" the following number.

5. What is $3.4\times 10^5$ written in standard decimal form?

   $340{,}000$. Move decimal 5 places right: $3.4 \rightarrow 340{,}000$.

6. What is $3.09\times10^4$ written in standard decimal form?

   $30900$. Move decimal 4 places right since exponent is positive.

7. Interpret technology output: if a calculator shows $3.2\text{E}7$, what is it in scientific notation?

   $3.2\times10^7$. E notation means "times 10 to the power of" the number after E.

8. Which is larger: $3.4\times 10^8$ or $5.6\times 10^7$?

   $3.4\times 10^8$. Compare exponents first: $10^8 > 10^7$.

9. What is $120000\div(3\times10^2)$ in scientific notation?

   $4\times10^2$. Convert $120000$ to $1.2\times10^5$, divide: $(1.2 \div 3) \times 10^{5-2} = 4 \times 10^2$.

10. What is $0.003\times(2\times10^4)$ in scientific notation?

    $6\times10^1$. Convert $0.003$ to $3×10^{-3}$, multiply: $(3×2)×10^{-3+4}=6×10^1$.

11. Identify the coefficient and exponent in $6.2\times10^{-5}$.

    Coefficient $6.2$; exponent $-5$. The coefficient is the decimal part; the exponent is the power of 10.

12. What is $\frac{6\times10^7}{2\times10^3}$ in scientific notation?

    $3\times10^4$. Divide coefficients $(6÷2=3)$, subtract exponents $(7-3=4)$.

13. What is $\frac{9\times10^{-4}}{3\times10^2}$ in scientific notation?

    $3\times10^{-6}$. Divide coefficients $(9÷3=3)$, subtract exponents $(-4-2=-6)$.

14. What is $6.1\times 10^{-3}$ written in standard decimal form?

    $0.0061$. Move decimal 3 places left: $6.1 \rightarrow 0.0061$.

15. What is the definition of scientific notation for a number?

    $a\times 10^n$ where $1\le |a|<10$ and $n$ is an integer. The coefficient must be between 1 and 10, with an integer exponent.

16. What is $(1.2\times 10^3)(5\times 10^2)$ in scientific notation?

    $6\times 10^5$. $(1.2×5)×10^{3+2} = 6×10^5$.

17. What is $(3\times10^5)(4\times10^2)$ in scientific notation?

    $1.2\times10^8$. Multiply coefficients $(3×4=12)$, add exponents $(5+2=7)$, adjust to proper form.

18. What is $9.1\times 10^{-3}$ written in standard decimal form?

    $0.0091$. Move decimal point 3 places left from 9.1.

19. What is $0.00045$ written in scientific notation?

    $4.5\times10^{-4}$. Move decimal 4 places right to get coefficient between 1 and 10.

20. State the rule for multiplying powers of $10$: $10^a\cdot10^b$ equals what?

    $10^{a+b}$. When multiplying powers with same base, add the exponents.

21. What is $(8\times10^{-6})(5\times10^3)$ in scientific notation?

    $4\times10^{-2}$. Multiply coefficients $(8×5=40)$, add exponents $(-6+3=-3)$, adjust to proper form.

22. What is the number $0.09\times 10^6$ written in proper scientific notation?

    $9\times 10^4$. Move decimal right: $0.09 = 9×10^{-2}$, so $0.09×10^6 = 9×10^4$.

23. What is the definition of scientific notation in standard form?

    $a\times 10^n$ where $1\le |a|<10$ and $n$ is an integer. The coefficient must be between 1 and 10, with an integer exponent.

24. Which unit is most appropriate for seafloor spreading at about $0.03\ \text{m/yr}$?

    $30\ \text{mm/yr}$. $0.03\ \text{m} = 30\ \text{mm}$, appropriate for small measurements.

25. What is the quotient $(8\times 10^7)\div(4\times 10^2)$ in scientific notation?

    $2\times 10^5$. Divide coefficients $(8÷4=2)$ and subtract exponents $(7-2=5)$.

26. What is $(4.5\times 10^6)+(2.0\times 10^6)$ in scientific notation?

    $6.5\times 10^6$. Same powers, so add coefficients: $4.5+2.0=6.5$.

27. What is $6.3\times 10^4$ written in standard decimal form?

    $63{,}000$. Move decimal point 4 places right from 6.3.

28. What is $(3.4\times10^5)-(1.9\times10^5)$ in scientific notation?

    $1.5\times10^5$. Subtract coefficients when exponents are same: $(3.4-1.9)×10^5$.

29. What is $7,200,000$ written in scientific notation?

    $7.2\times10^6$. Move decimal 6 places left to get coefficient between 1 and 10.

30. What is $6.0\times 10^3+2.5\times 10^2$ written in scientific notation?

    $6.25\times 10^3$. $2.5×10^2 = 0.25×10^3$, then $6.0+0.25=6.25$.

31. What is $(9\times 10^3)(7\times 10^{-2})$ in scientific notation?

    $6.3\times 10^2$. $(9×7)×10^{3+(-2)} = 63×10^1 = 6.3×10^2$.

32. What is $4.5\times10^6+2.0\times10^5$ in scientific notation?

    $4.7\times10^6$. Convert to same exponent: $4.5×10^6+0.2×10^6=4.7×10^6$.

33. What does a calculator display of $3.1\text{E}6$ mean in scientific notation?

    $3.1\times 10^6$. E notation means "times 10 to the power of".

34. What is $0.00072$ written in scientific notation?

    $7.2\times 10^{-4}$. Move decimal 4 places right to get 7.2, so multiply by $10^{-4}$.

35. What is $(3\times 10^8)(4\times 10^{-3})$ in scientific notation?

    $1.2\times 10^6$. $(3×4)×10^{8+(-3)} = 12×10^5 = 1.2×10^6$.

36. What is $6.02\times 10^{23}$ rounded to $2$ significant figures?

    $6.0\times 10^{23}$. Round 6.02 to 2 sig figs gives 6.0, keep the exponent.

37. What is $(7.5\times 10^{-6})\div(2.5\times 10^{-2})$ in scientific notation?

    $3\times 10^{-4}$. $(7.5÷2.5)×10^{-6-(-2)} = 3×10^{-4}$.

38. What is $(0.0009)\div(3\times 10^{-4})$ in scientific notation?

    $3\times 10^0$. $(9×10^{-4})÷(3×10^{-4}) = 3×10^0 = 3$.

39. Identify the exponent in $7.3\times 10^{-5}$ and state what it means about size.

    $-5$; the value is less than $1$. Negative exponent means the decimal point moves left.

40. What is $(7.2\times 10^5)-(3.2\times 10^5)$ in scientific notation?

    $4.0\times 10^5$. Same powers, so subtract coefficients: $7.2-3.2=4.0$.

41. State the rule for dividing powers of $10$: $\frac{10^a}{10^b}$ equals what?

    $10^{a-b}$. When dividing powers with same base, subtract the exponents.

42. What is $(8\times 10^7)\div(2\times 10^3)$ in scientific notation?

    $4\times 10^4$. Divide coefficients $(8÷2=4)$ and subtract exponents $(7-3=4)$.

43. What is $0.004\times 10^6$ written in scientific notation?

    $4\times 10^3$. $0.004 = 4×10^{-3}$, so $4×10^{-3}×10^6 = 4×10^3$.

44. What is $5.6\times10^{-3}$ written in standard decimal form?

    $0.0056$. Move decimal 3 places left since exponent is negative.

45. What is $3.6\times 10^7-9\times 10^6$ written in scientific notation?

    $2.7\times 10^7$. $9×10^6 = 0.9×10^7$, then $3.6-0.9=2.7$.

46. Interpret technology output: if a calculator shows $-5.6\text{E}-3$, what is it in scientific notation?

    $-5.6\times10^{-3}$. E notation with negative exponent indicates a small decimal value.

47. What power of $10$ is used when you move a decimal point left $k$ places?

    Multiply by $10^k$. Moving left increases the exponent by the number of places moved.

48. What is $(5\times 10^6)+(2\times 10^6)$ in scientific notation?

    $7\times 10^6$. Same powers, so add coefficients: $(5+2)×10^6$.

49. What is $(6.0\times10^3)+(2.0\times10^3)$ in scientific notation?

    $8.0\times10^3$. Add coefficients when exponents are same: $(6.0+2.0)×10^3$.

50. What power of $10$ is used when you move a decimal point right $k$ places?

    Multiply by $10^{-k}$. Moving right decreases the exponent by the number of places moved.

51. What is $4{,}500{,}000$ written in scientific notation?

    $4.5\times 10^6$. Move decimal 6 places left, so exponent is positive 6.

52. What is $3.2\times 10^4+6.8\times 10^3$ in scientific notation?

    $3.88\times 10^4$. Convert to same power: $3.2×10^4 + 0.68×10^4 = 3.88×10^4$.

53. What is $(4\times 10^5)-(1.5\times 10^5)$ in scientific notation?

    $2.5\times 10^5$. Same powers, so subtract coefficients: $(4-1.5)×10^5$.

54. What is $(2\times 10^4)(3\times 10^2)$ in scientific notation?

    $6\times 10^6$. Multiply coefficients $(2×3=6)$ and add exponents $(4+2=6)$.

55. What is $(9\times 10^{-2})\div(3\times 10^4)$ in scientific notation?

    $3\times 10^{-6}$. $(9÷3)×10^{-2-4} = 3×10^{-6}$.

56. Which unit is most appropriate for a thickness of $0.000003\text{ m}$: $\text{m}$, $\text{mm}$, or $\mu\text{m}$?

    $\mu\text{m}$. $0.000003\text{ m} = 3\mu\text{m}$, avoiding many zeros.

57. What is $5{,}600{,}000$ written in scientific notation?

    $5.6\times 10^6$. Move decimal 6 places left to get 5.6, so multiply by $10^6$.

58. What is the product $(3\times 10^5)(2\times 10^3)$ in scientific notation?

    $6\times 10^8$. Multiply coefficients $(3×2=6)$ and add exponents $(5+3=8)$.

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