HighSchoolQ

➕ Algebra 1 · Exponents and Polynomials

Scientific Notation: notes and common mistakes

Study notes

Key ideas

  • Scientific notation is a × 10n with 1 ≤ a < 10 and n an integer. All numbers here are positive.
  • The exponent counts how many places the decimal point slides. Sliding left (a number of 10 or more) gives a positive exponent. Sliding right (a number between 0 and 1) gives a negative exponent.
  • For positive numbers in scientific notation, compare the exponents first. If they are equal, compare the front numbers. To find how many times as large, divide larger by smaller.
  • Multiply: multiply the fronts and add the exponents. Divide: divide the fronts and subtract the exponents. Then fix the front number if it left the range from 1 up to (but not including) 10.
  • Add or subtract: match the powers of 10 first, then add or subtract the fronts and keep the power.

Steps

  1. To convert: slide the decimal point until one nonzero digit is on its left, count the slides, and set the sign by the direction.
  2. To multiply or divide: work on the front numbers and on the exponents separately.
  3. To add or subtract: rewrite one number so both use the same power of 10.
  4. Fix the front number so it is at least 1 and less than 10.
  5. Check with a size estimate or with ordinary numbers, and keep the units straight.

Formulas

  • Scientific notation: a × 10n with 1 ≤ a < 10 and n an integer
  • Multiply: (a × 10m)(b × 10n) = (a × b) × 10m+n
  • Divide: (a × 10m) ÷ (b × 10n) = (a ÷ b) × 10m−n
  • Add: a × 10n + b × 10n = (a + b) × 10n (same exponent)
  • Subtract: a × 10n − b × 10n = (a − b) × 10n (same exponent)
  • How many times as large: (larger number) ÷ (smaller number)

Watch out

  • Count the slides of the decimal point, not the zeros: 1,700,000 has five zeros, but it equals 1.7 × 106.
  • A negative exponent does not make a negative number: 2 × 10−3 = 0.002, a small positive number.
  • When multiplying, add the exponents (do not multiply them). When adding, do not add the exponents: match the powers, then add the front numbers.
  • After multiplying or dividing, check the front number. 22 × 104 and 0.4 × 104 are not finished.
  • Subtracting a negative exponent adds: 5 − (−3) = 8, not 2.
  • Check the units: convert so they fit together before you add, subtract, multiply or divide.

Why it works

  • Why does the exponent count the slides? Each slide of the decimal point changes the front number by a factor of 10, and the power of 10 undoes it, so the value stays the same.
  • Why add exponents when multiplying? Reorder the factors so the two powers of 10 sit together. 10m × 10n uses m factors of 10 and then n more factors of 10, which is m + n factors of 10 in all, and the product rule keeps working when an exponent is negative.
  • Why match the powers before adding? It is the same as combining like terms. 3x + 5x = 8x works because both terms count the same unit (x). In the same way, 3 × 105 + 5 × 105 = 8 × 105 counts the same unit (105).

Common mistakes

✗ MistakeThe exponent is the number of zeros: 1,700,000 = 1.7 × 105.
✓ InsteadCount the places the decimal point slides, not the zeros. 1,700,000 → 1.7 slides 6 places, so 1,700,000 = 1.7 × 106.
✗ MistakeA negative exponent makes the number negative: 8 × 10−4 = −8,000.
✓ InsteadA negative exponent means dividing by a power of 10: 8 × 10−4 = 8 ÷ 10,000 = 0.0008, a small positive number.
✗ MistakeTo add, add the exponents: 3 × 102 + 4 × 102 = 7 × 104.
✓ InsteadWhen the powers match, add the front numbers and keep the power: 3 × 102 + 4 × 102 = 7 × 102. Only multiplying adds exponents.
✗ MistakeTo multiply, multiply the exponents: (2 × 103)(4 × 105) = 8 × 1015.
✓ InsteadMultiply the front numbers and add the exponents: (2 × 103)(4 × 105) = 8 × 108.
✗ Mistake38 × 106 is fine, because the value is right.
✓ InsteadThe front number must be at least 1 and less than 10. Slide the point left and raise the exponent: 38 × 106 = 3.8 × 107.

Teach it back

Explain to a friend how to write 0.0000725 in scientific notation, why its exponent is negative, and how to multiply it by 4 × 103.

Explaining it out loud shows what you really know. Did you…

  • I slid the point 5 places right to get 7.25 and wrote 7.25 × 10−5.
  • I said the exponent is negative because sliding right made the front number 105 times bigger, so the power of 10 has to divide it back. The number itself is still positive.
  • I multiplied the fronts (7.25 × 4 = 29) and added the exponents (−5 + 3 = −2), then fixed the front number: 29 × 10−2 = 2.9 × 10−1.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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