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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.

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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.

Κάρτες σε αυτήν τη δέσμη

  1. Κάρτα 1

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 2

    Ερώτηση

    Exact arithmetic: write 680000680000 in normalized scientific notation.

    Απάντηση

    6.8×1056.8\times10^5

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

  3. Κάρτα 3

    Ερώτηση

    Exact arithmetic: evaluate 10010^0.

    Απάντηση

    11

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

  4. Κάρτα 4

    Ερώτηση

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

    Απάντηση

    0.000730.00073

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

  5. Κάρτα 5

    Ερώτηση

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

    Απάντηση

    4.82×1044.82\times10^4

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

  6. Κάρτα 6

    Ερώτηση

    Exact arithmetic: write 0.00009160.0000916 in normalized scientific notation.

    Απάντηση

    9.16×10−59.16\times10^{-5}

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

  7. Κάρτα 7

    Ερώτηση

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

    Απάντηση

    0.00010.0001

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

  8. Κάρτα 8

    Ερώτηση

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

    Απάντηση

    20640002064000

    Move the decimal 6 places right.

  9. Κάρτα 9

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 10

    Ερώτηση

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

    Απάντηση

    −4.08×10−3-4.08\times10^{-3}

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

  11. Κάρτα 11

    Ερώτηση

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

    Απάντηση

    −820-820

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

  12. Κάρτα 12

    Ερώτηση

    Exact arithmetic: write 5.045.04 in normalized scientific notation.

    Απάντηση

    5.04×1005.04\times10^0

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

  13. Κάρτα 13

    Ερώτηση

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

    Απάντηση

    −7.2×10−3-7.2\times10^{-3}

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

  14. Κάρτα 14

    Ερώτηση

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

    Απάντηση

    −0.0315-0.0315

    Divide 3.15 by 100 and keep the negative sign.

  15. Κάρτα 15

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 16

    Ερώτηση

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

    Απάντηση

    No. Write zero as 00.

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

  17. Κάρτα 17

    Ερώτηση

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

    Απάντηση

    5.6×10−55.6\times10^{-5}

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

  18. Κάρτα 18

    Ερώτηση

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

    Απάντηση

    3×1053\times10^5

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

  19. Κάρτα 19

    Ερώτηση

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

    Απάντηση

    5.58×1045.58\times10^4

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

  20. Κάρτα 20

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 21

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 22

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 23

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 24

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 25

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 26

    Ερώτηση

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

    Απάντηση

    −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. Κάρτα 27

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 28

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 29

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 30

    Ερώτηση

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

    Απάντηση

    −9.2×10−1-9.2\times10^{-1}

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

  31. Κάρτα 31

    Ερώτηση

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

    Απάντηση

    6×1006\times10^0

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

  32. Κάρτα 32

    Ερώτηση

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

    Απάντηση

    −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. Κάρτα 33

    Ερώτηση

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

    Απάντηση

    −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. Κάρτα 34

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 35

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 36

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 37

    Ερώτηση

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

    Απάντηση

    4×1074\times10^7

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

  38. Κάρτα 38

    Ερώτηση

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

    Απάντηση

    00

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

  39. Κάρτα 39

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 40

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 41

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 42

    Ερώτηση

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

    Απάντηση

    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 κάρτες

Scientific Notation Flashcards: Conversions, Operations & Error Checks

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