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Scientific Notation Flashcards: Conversions, Operations & Error Checks

Practice exact scientific notation with 42 solved cards on decimal conversions, normalization, all four operations, signed values, and local error checks.

By author Open Study Contributor

About this deck

Practice scientific notation through 42 short cards on exact decimal conversions, value-preserving normalization, multiplication, division, addition, subtraction and error checks. Numerical prompts explicitly use exact arithmetic: no measurement rounding or significant-figure rule is implied. Answers start with the result, followed by a brief reason or calculation.

The deck practices ordinary decimal → normalized scientific notation and scientific notation → ordinary decimal with different examples, plus nonnormalized expression → equivalent normalized expression, two-term operation → result, and erroneous step → correction. Signed coefficients, negative and zero exponents, the coefficient boundaries and exact cancellation to zero are included. For a nonzero normalized result, the coefficient has absolute value at least 1 and less than 10; zero is written as 0.

The sequence introduces the notation and conversions before operations, then mixes operations and local repairs. Related variants are separated by at least three other cards. All cards use the tag scientific-notation-practice. Decimal arithmetic is assumed. There are no automatic reverse copies: conversion directions use distinct inputs, and an arithmetic result does not uniquely determine its original operation. The deck excludes measurement uncertainty, significant-figure rounding, units and metric prefixes, engineering notation, logarithms and exam preparation. It is a focused practice set rather than a complete mathematics course.

Questions, answers, explanations, organization and metadata were independently written with AI assistance. Common mathematical facts were checked against NASA: Powers of Numbers and the notation convention in NIST SI Guide, Appendix B.2. No source exercises, wording, diagrams or physical constants were copied. The cover is an original AI-generated illustration; the cards contain no images.

CC0 applies to the original expression and generated cover to the extent applicable rights exist. It does not claim ownership of mathematical facts or rights in third-party material. This independent resource is not affiliated with or endorsed by NASA, NIST or an examination provider.

Cards in this deck

  1. Card 1

    Question

    For a nonzero number in normalized scientific notation a×10na\times10^n, what restrictions apply to aa and nn?

    Answer

    1≤∣a∣<101\leq |a|<10, and nn is an integer.

    The coefficient may be negative; its absolute value must be at least 1 and less than 10.

  2. Card 2

    Question

    Exact arithmetic: write 680000680000 in normalized scientific notation.

    Answer

    6.8×1056.8\times10^5

    The decimal moves 5 places left; the factor 10510^5 preserves the value.

  3. Card 3

    Question

    Exact arithmetic: evaluate 10010^0.

    Answer

    11

    A zero exponent does not make a nonzero base equal zero.

  4. Card 4

    Question

    Exact arithmetic: write 7.3×10−47.3\times10^{-4} as an ordinary decimal.

    Answer

    0.000730.00073

    Multiplying by 10−410^{-4} divides 7.3 by 10000.

  5. Card 5

    Question

    Exact arithmetic: normalize 48.2×10348.2\times10^3 without changing its value.

    Answer

    4.82×1044.82\times10^4

    Dividing the coefficient by 10 requires multiplying the power-of-ten factor by 10.

  6. Card 6

    Question

    Exact arithmetic: write 0.00009160.0000916 in normalized scientific notation.

    Answer

    9.16×10−59.16\times10^{-5}

    The decimal moves 5 places right to make the coefficient; the exponent is negative.

  7. Card 7

    Question

    Exact arithmetic: evaluate 10−410^{-4} as an ordinary decimal.

    Answer

    0.00010.0001

    The negative exponent means a reciprocal: 10−4=1/10410^{-4}=1/10^4. The value is positive.

  8. Card 8

    Question

    Exact arithmetic: write 2.064×1062.064\times10^6 as an ordinary decimal.

    Answer

    20640002064000

    Move the decimal 6 places right.

  9. Card 9

    Question

    Exact arithmetic: normalize 0.063×1050.063\times10^5 without changing its value.

    Answer

    6.3×1036.3\times10^3

    The coefficient increases by a factor of 100, so the power-of-ten factor decreases by a factor of 100.

  10. Card 10

    Question

    Exact arithmetic: write −0.00408-0.00408 in normalized scientific notation.

    Answer

    −4.08×10−3-4.08\times10^{-3}

    Keep the negative sign on the coefficient. The exponent describes the decimal scale.

  11. Card 11

    Question

    Exact arithmetic: write −8.2×102-8.2\times10^2 as an ordinary decimal.

    Answer

    −820-820

    Multiplying by 100 changes the size, while the negative sign remains.

  12. Card 12

    Question

    Exact arithmetic: write 5.045.04 in normalized scientific notation.

    Answer

    5.04×1005.04\times10^0

    The coefficient is already in the required range, and 100=110^0=1.

  13. Card 13

    Question

    Exact arithmetic: normalize −720×10−5-720\times10^{-5} without changing its value.

    Answer

    −7.2×10−3-7.2\times10^{-3}

    Divide the coefficient by 100 and increase the exponent by 2.

  14. Card 14

    Question

    Exact arithmetic: write −3.15×10−2-3.15\times10^{-2} as an ordinary decimal.

    Answer

    −0.0315-0.0315

    Divide 3.15 by 100 and keep the negative sign.

  15. Card 15

    Question

    Exact arithmetic: calculate (2.4×103)(3×104)(2.4\times10^3)(3\times10^4). Give normalized scientific notation.

    Answer

    7.2×1077.2\times10^7

    Multiply the coefficients: 2.4×3=7.22.4\times3=7.2. Add the exponents: 3+4=73+4=7.

  16. Card 16

    Question

    Can zero satisfy the normalized scientific-notation condition 1≤∣a∣<101\leq|a|<10?

    Answer

    No. Write zero as 00.

    A coefficient satisfying that condition cannot produce zero when multiplied by a power of ten.

  17. Card 17

    Question

    Exact arithmetic: normalize 0.0056×10−20.0056\times10^{-2} without changing its value.

    Answer

    5.6×10−55.6\times10^{-5}

    Multiply the coefficient by 1000 and decrease the exponent by 3.

  18. Card 18

    Question

    Exact arithmetic: calculate (8.1×108)/(2.7×103)(8.1\times10^8)/(2.7\times10^3). Give normalized scientific notation.

    Answer

    3×1053\times10^5

    Divide the coefficients and subtract the denominator exponent: 8−3=58-3=5.

  19. Card 19

    Question

    Exact arithmetic: calculate (3.16×104)+(2.42×104)(3.16\times10^4)+(2.42\times10^4). Give normalized scientific notation.

    Answer

    5.58×1045.58\times10^4

    The powers match, so add the coefficients and retain 10410^4.

  20. Card 20

    Question

    Exact arithmetic: calculate (6.4×10−3)(5×102)(6.4\times10^{-3})(5\times10^2). Give normalized scientific notation.

    Answer

    3.2×1003.2\times10^0

    The product is 32×10−132\times10^{-1}. Normalize by dividing 32 by 10 and increasing the exponent by 1.

  21. Card 21

    Question

    Exact arithmetic: calculate (7.85×10−6)−(2.65×10−6)(7.85\times10^{-6})-(2.65\times10^{-6}). Give normalized scientific notation.

    Answer

    5.2×10−65.2\times10^{-6}

    The powers match; subtract the coefficients: 7.85−2.65=5.27.85-2.65=5.2.

  22. Card 22

    Question

    Exact arithmetic: normalize 10×10−710\times10^{-7} without changing its value.

    Answer

    1×10−61\times10^{-6}

    A coefficient of 10 is outside the normalized range. Divide it by 10 and increase the exponent by 1.

  23. Card 23

    Question

    Exact arithmetic: calculate (1.8×105)/(6×102)(1.8\times10^5)/(6\times10^2). Give normalized scientific notation.

    Answer

    3×1023\times10^2

    The quotient is 0.3×1030.3\times10^3. Multiply the coefficient by 10 and decrease the exponent by 1.

  24. Card 24

    Question

    Exact arithmetic: calculate (4.2×105)+(3.6×104)(4.2\times10^5)+(3.6\times10^4). Give normalized scientific notation.

    Answer

    4.56×1054.56\times10^5

    Rewrite 3.6×1043.6\times10^4 as 0.36×1050.36\times10^5, then add 4.2+0.364.2+0.36.

  25. Card 25

    Question

    Exact arithmetic: repair the exponent: 0.00052=5.2×1040.00052=5.2\times10^4.

    Answer

    0.00052=5.2×10−40.00052=5.2\times10^{-4}

    The number is smaller than 1. A positive exponent here would give 52000.

  26. Card 26

    Question

    Exact arithmetic: calculate (−1.6×10−4)(2.5×10−2)(-1.6\times10^{-4})(2.5\times10^{-2}). Give normalized scientific notation.

    Answer

    −4×10−6-4\times10^{-6}

    The signs give a negative product; 1.6×2.5=41.6\times2.5=4 and −4+(−2)=−6-4+(-2)=-6.

  27. Card 27

    Question

    Exact arithmetic: calculate (6.3×103)−(8×102)(6.3\times10^3)-(8\times10^2). Give normalized scientific notation.

    Answer

    5.5×1035.5\times10^3

    Rewrite 8×1028\times10^2 as 0.8×1030.8\times10^3, then subtract 6.3−0.86.3-0.8.

  28. Card 28

    Question

    Exact arithmetic: calculate (7.2×10−5)/(2.4×10−2)(7.2\times10^{-5})/(2.4\times10^{-2}). Give normalized scientific notation.

    Answer

    3×10−33\times10^{-3}

    Divide the coefficients: 7.2/2.4=37.2/2.4=3. The exponent is −5−(−2)=−3-5-(-2)=-3.

  29. Card 29

    Question

    Exact arithmetic: calculate (7.8×106)+(6.5×106)(7.8\times10^6)+(6.5\times10^6). Give normalized scientific notation.

    Answer

    1.43×1071.43\times10^7

    The sum is 14.3×10614.3\times10^6; normalize the coefficient by moving one factor of 10 into the power.

  30. Card 30

    Question

    Exact arithmetic: normalize −0.92×100-0.92\times10^0 without changing its value.

    Answer

    −9.2×10−1-9.2\times10^{-1}

    Increase the coefficient magnitude by a factor of 10 and decrease the exponent by 1.

  31. Card 31

    Question

    Exact arithmetic: calculate (4×10−7)(1.5×107)(4\times10^{-7})(1.5\times10^7). Give normalized scientific notation.

    Answer

    6×1006\times10^0

    The coefficient product is 6. The exponents add to zero, so the power-of-ten factor is 1.

  32. Card 32

    Question

    Exact arithmetic: calculate (2.1×10−3)−(8.4×10−3)(2.1\times10^{-3})-(8.4\times10^{-3}). Give normalized scientific notation.

    Answer

    −6.3×10−3-6.3\times10^{-3}

    The powers match, and 2.1−8.4=−6.32.1-8.4=-6.3. A negative coefficient is allowed.

  33. Card 33

    Question

    Exact arithmetic: calculate (6.3×10−2)/(−9×10−4)(6.3\times10^{-2})/(-9\times10^{-4}). Give normalized scientific notation.

    Answer

    −7×101-7\times10^1

    The quotient is −0.7×102-0.7\times10^2 because −2−(−4)=2-2-(-4)=2. Normalize to −7×101-7\times10^1.

  34. Card 34

    Question

    Exact arithmetic: repair the final expression: 37×104=3.7×10337\times10^4=3.7\times10^3.

    Answer

    37×104=3.7×10537\times10^4=3.7\times10^5

    Dividing the coefficient by 10 must be offset by increasing the exponent by 1.

  35. Card 35

    Question

    Exact arithmetic: calculate (−5.2×10−4)+(1.7×10−3)(-5.2\times10^{-4})+(1.7\times10^{-3}). Give normalized scientific notation.

    Answer

    1.18×10−31.18\times10^{-3}

    Use a common power: −0.52×10−3+1.7×10−3-0.52\times10^{-3}+1.7\times10^{-3}. Add the signed coefficients.

  36. Card 36

    Question

    Exact arithmetic: calculate (−3.5×102)(−2×10−5)(-3.5\times10^2)(-2\times10^{-5}). Give normalized scientific notation.

    Answer

    7×10−37\times10^{-3}

    Two negative factors give a positive product. Multiply 3.5×23.5\times2 and add 2+(−5)2+(-5).

  37. Card 37

    Question

    Exact arithmetic: calculate (4.8×103)/(1.2×10−4)(4.8\times10^3)/(1.2\times10^{-4}). Give normalized scientific notation.

    Answer

    4×1074\times10^7

    The coefficient quotient is 4. Subtract the negative denominator exponent: 3−(−4)=73-(-4)=7.

  38. Card 38

    Question

    Exact arithmetic: calculate (4.6×109)−(4.6×109)(4.6\times10^9)-(4.6\times10^9).

    Answer

    00

    Identical values cancel exactly. Zero has no representation satisfying 1≤∣a∣<101\leq|a|<10.

  39. Card 39

    Question

    Exact arithmetic: repair the result: (3×104)+(2×103)=5×107(3\times10^4)+(2\times10^3)=5\times10^7.

    Answer

    3.2×1043.2\times10^4

    Rewrite the second term as 0.2×1040.2\times10^4 and add 3+0.23+0.2. Adding exponents is a multiplication rule, not an addition rule.

  40. Card 40

    Question

    Exact arithmetic: calculate (8×10−6)(7×10−3)(8\times10^{-6})(7\times10^{-3}). Give normalized scientific notation.

    Answer

    5.6×10−85.6\times10^{-8}

    The product is 56×10−956\times10^{-9}. Dividing the coefficient by 10 raises the exponent to −8-8.

  41. Card 41

    Question

    Exact arithmetic: calculate (2.1×10−5)/(7×10−5)(2.1\times10^{-5})/(7\times10^{-5}). Give normalized scientific notation.

    Answer

    3×10−13\times10^{-1}

    Equal powers cancel, leaving 0.3×1000.3\times10^0. Normalize the coefficient to 3 and decrease the exponent by 1.

  42. Card 42

    Question

    Exact arithmetic: repair the exponent: (9×102)/(3×105)=3×103(9\times10^2)/(3\times10^5)=3\times10^3.

    Answer

    3×10−33\times10^{-3}

    Division subtracts the denominator exponent from the numerator exponent: 2−5=−32-5=-3.

Navy ten with a superscript n above three small coral and teal tiles on ivory paper.

42 cards

Scientific Notation Flashcards: Conversions, Operations & Error Checks

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