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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.

ಲೇಖಕ Open Study Contributor

ಈ ಡೆಕ್ ಬಗ್ಗೆ

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.

ಈ ಡೆಕ್‌ನ ಕಾರ್ಡ್‌ಗಳು

  1. ಕಾರ್ಡ್ 1

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 2

    ಪ್ರಶ್ನೆ

    Exact arithmetic: write 680000680000 in normalized scientific notation.

    ಉತ್ತರ

    6.8×1056.8\times10^5

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

  3. ಕಾರ್ಡ್ 3

    ಪ್ರಶ್ನೆ

    Exact arithmetic: evaluate 10010^0.

    ಉತ್ತರ

    11

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

  4. ಕಾರ್ಡ್ 4

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    0.000730.00073

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

  5. ಕಾರ್ಡ್ 5

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    4.82×1044.82\times10^4

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

  6. ಕಾರ್ಡ್ 6

    ಪ್ರಶ್ನೆ

    Exact arithmetic: write 0.00009160.0000916 in normalized scientific notation.

    ಉತ್ತರ

    9.16×10−59.16\times10^{-5}

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

  7. ಕಾರ್ಡ್ 7

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    0.00010.0001

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

  8. ಕಾರ್ಡ್ 8

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    20640002064000

    Move the decimal 6 places right.

  9. ಕಾರ್ಡ್ 9

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 10

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    −4.08×10−3-4.08\times10^{-3}

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

  11. ಕಾರ್ಡ್ 11

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    −820-820

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

  12. ಕಾರ್ಡ್ 12

    ಪ್ರಶ್ನೆ

    Exact arithmetic: write 5.045.04 in normalized scientific notation.

    ಉತ್ತರ

    5.04×1005.04\times10^0

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

  13. ಕಾರ್ಡ್ 13

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    −7.2×10−3-7.2\times10^{-3}

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

  14. ಕಾರ್ಡ್ 14

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    −0.0315-0.0315

    Divide 3.15 by 100 and keep the negative sign.

  15. ಕಾರ್ಡ್ 15

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 16

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    No. Write zero as 00.

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

  17. ಕಾರ್ಡ್ 17

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    5.6×10−55.6\times10^{-5}

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

  18. ಕಾರ್ಡ್ 18

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    3×1053\times10^5

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

  19. ಕಾರ್ಡ್ 19

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    5.58×1045.58\times10^4

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

  20. ಕಾರ್ಡ್ 20

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 21

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 22

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 23

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 24

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 25

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 26

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    −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. ಕಾರ್ಡ್ 27

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 28

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 29

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 30

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    −9.2×10−1-9.2\times10^{-1}

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

  31. ಕಾರ್ಡ್ 31

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    6×1006\times10^0

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

  32. ಕಾರ್ಡ್ 32

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    −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. ಕಾರ್ಡ್ 33

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    −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. ಕಾರ್ಡ್ 34

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 35

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 36

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 37

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    4×1074\times10^7

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

  38. ಕಾರ್ಡ್ 38

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    00

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

  39. ಕಾರ್ಡ್ 39

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 40

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 41

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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. ಕಾರ್ಡ್ 42

    ಪ್ರಶ್ನೆ

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

    ಉತ್ತರ

    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 ಕಾರ್ಡ್‌ಗಳು

Scientific Notation Flashcards: Conversions, Operations & Error Checks

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