باز کردن برنامه

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.

نوشتهٔ Open Study Contributor

دربارهٔ این دسته

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.

کارت‌های این دسته

  1. کارت ۱

    پرسش

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

    پاسخ

    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. کارت ۲

    پرسش

    Exact arithmetic: write 680000680000 in normalized scientific notation.

    پاسخ

    6.8×1056.8\times10^5

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

  3. کارت ۳

    پرسش

    Exact arithmetic: evaluate 10010^0.

    پاسخ

    11

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

  4. کارت ۴

    پرسش

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

    پاسخ

    0.000730.00073

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

  5. کارت ۵

    پرسش

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

    پاسخ

    4.82×1044.82\times10^4

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

  6. کارت ۶

    پرسش

    Exact arithmetic: write 0.00009160.0000916 in normalized scientific notation.

    پاسخ

    9.16×10−59.16\times10^{-5}

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

  7. کارت ۷

    پرسش

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

    پاسخ

    0.00010.0001

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

  8. کارت ۸

    پرسش

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

    پاسخ

    20640002064000

    Move the decimal 6 places right.

  9. کارت ۹

    پرسش

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

    پاسخ

    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. کارت ۱۰

    پرسش

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

    پاسخ

    −4.08×10−3-4.08\times10^{-3}

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

  11. کارت ۱۱

    پرسش

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

    پاسخ

    −820-820

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

  12. کارت ۱۲

    پرسش

    Exact arithmetic: write 5.045.04 in normalized scientific notation.

    پاسخ

    5.04×1005.04\times10^0

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

  13. کارت ۱۳

    پرسش

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

    پاسخ

    −7.2×10−3-7.2\times10^{-3}

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

  14. کارت ۱۴

    پرسش

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

    پاسخ

    −0.0315-0.0315

    Divide 3.15 by 100 and keep the negative sign.

  15. کارت ۱۵

    پرسش

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

    پاسخ

    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. کارت ۱۶

    پرسش

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

    پاسخ

    No. Write zero as 00.

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

  17. کارت ۱۷

    پرسش

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

    پاسخ

    5.6×10−55.6\times10^{-5}

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

  18. کارت ۱۸

    پرسش

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

    پاسخ

    3×1053\times10^5

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

  19. کارت ۱۹

    پرسش

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

    پاسخ

    5.58×1045.58\times10^4

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

  20. کارت ۲۰

    پرسش

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

    پاسخ

    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. کارت ۲۱

    پرسش

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

    پاسخ

    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. کارت ۲۲

    پرسش

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

    پاسخ

    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. کارت ۲۳

    پرسش

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

    پاسخ

    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. کارت ۲۴

    پرسش

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

    پاسخ

    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. کارت ۲۵

    پرسش

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

    پاسخ

    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. کارت ۲۶

    پرسش

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

    پاسخ

    −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. کارت ۲۷

    پرسش

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

    پاسخ

    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. کارت ۲۸

    پرسش

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

    پاسخ

    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. کارت ۲۹

    پرسش

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

    پاسخ

    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. کارت ۳۰

    پرسش

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

    پاسخ

    −9.2×10−1-9.2\times10^{-1}

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

  31. کارت ۳۱

    پرسش

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

    پاسخ

    6×1006\times10^0

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

  32. کارت ۳۲

    پرسش

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

    پاسخ

    −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. کارت ۳۳

    پرسش

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

    پاسخ

    −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. کارت ۳۴

    پرسش

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

    پاسخ

    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. کارت ۳۵

    پرسش

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

    پاسخ

    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. کارت ۳۶

    پرسش

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

    پاسخ

    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. کارت ۳۷

    پرسش

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

    پاسخ

    4×1074\times10^7

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

  38. کارت ۳۸

    پرسش

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

    پاسخ

    00

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

  39. کارت ۳۹

    پرسش

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

    پاسخ

    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. کارت ۴۰

    پرسش

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

    پاسخ

    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. کارت ۴۱

    پرسش

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

    پاسخ

    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. کارت ۴۲

    پرسش

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

    پاسخ

    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.

۴۲ کارت

Scientific Notation Flashcards: Conversions, Operations & Error Checks

این دسته را رایگان مطالعه کنید

Nibomo باز می‌شود تا بتوانید مطالعه را شروع کنید.