Відкрити застосунок

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.

Автор Open Study Contributor

Про цю колоду

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.

Картки в цій колоді

  1. Картка 1

    Питання

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

    Відповідь

    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. Картка 2

    Питання

    Exact arithmetic: write 680000680000 in normalized scientific notation.

    Відповідь

    6.8×1056.8\times10^5

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

  3. Картка 3

    Питання

    Exact arithmetic: evaluate 10010^0.

    Відповідь

    11

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

  4. Картка 4

    Питання

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

    Відповідь

    0.000730.00073

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

  5. Картка 5

    Питання

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

    Відповідь

    4.82×1044.82\times10^4

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

  6. Картка 6

    Питання

    Exact arithmetic: write 0.00009160.0000916 in normalized scientific notation.

    Відповідь

    9.16×10−59.16\times10^{-5}

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

  7. Картка 7

    Питання

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

    Відповідь

    0.00010.0001

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

  8. Картка 8

    Питання

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

    Відповідь

    20640002064000

    Move the decimal 6 places right.

  9. Картка 9

    Питання

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

    Відповідь

    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. Картка 10

    Питання

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

    Відповідь

    −4.08×10−3-4.08\times10^{-3}

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

  11. Картка 11

    Питання

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

    Відповідь

    −820-820

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

  12. Картка 12

    Питання

    Exact arithmetic: write 5.045.04 in normalized scientific notation.

    Відповідь

    5.04×1005.04\times10^0

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

  13. Картка 13

    Питання

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

    Відповідь

    −7.2×10−3-7.2\times10^{-3}

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

  14. Картка 14

    Питання

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

    Відповідь

    −0.0315-0.0315

    Divide 3.15 by 100 and keep the negative sign.

  15. Картка 15

    Питання

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

    Відповідь

    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. Картка 16

    Питання

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

    Відповідь

    No. Write zero as 00.

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

  17. Картка 17

    Питання

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

    Відповідь

    5.6×10−55.6\times10^{-5}

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

  18. Картка 18

    Питання

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

    Відповідь

    3×1053\times10^5

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

  19. Картка 19

    Питання

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

    Відповідь

    5.58×1045.58\times10^4

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

  20. Картка 20

    Питання

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

    Відповідь

    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. Картка 21

    Питання

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

    Відповідь

    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. Картка 22

    Питання

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

    Відповідь

    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. Картка 23

    Питання

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

    Відповідь

    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. Картка 24

    Питання

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

    Відповідь

    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. Картка 25

    Питання

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

    Відповідь

    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. Картка 26

    Питання

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

    Відповідь

    −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. Картка 27

    Питання

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

    Відповідь

    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. Картка 28

    Питання

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

    Відповідь

    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. Картка 29

    Питання

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

    Відповідь

    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. Картка 30

    Питання

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

    Відповідь

    −9.2×10−1-9.2\times10^{-1}

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

  31. Картка 31

    Питання

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

    Відповідь

    6×1006\times10^0

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

  32. Картка 32

    Питання

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

    Відповідь

    −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. Картка 33

    Питання

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

    Відповідь

    −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. Картка 34

    Питання

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

    Відповідь

    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. Картка 35

    Питання

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

    Відповідь

    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. Картка 36

    Питання

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

    Відповідь

    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. Картка 37

    Питання

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

    Відповідь

    4×1074\times10^7

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

  38. Картка 38

    Питання

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

    Відповідь

    00

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

  39. Картка 39

    Питання

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

    Відповідь

    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. Картка 40

    Питання

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

    Відповідь

    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. Картка 41

    Питання

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

    Відповідь

    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. Картка 42

    Питання

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

    Відповідь

    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 картки

Scientific Notation Flashcards: Conversions, Operations & Error Checks

Вивчати цю колоду безкоштовно

Відкриється Nibomo, і ви зможете почати навчання.