HighSchoolQ

➕ Algebra 1 · Exponents and Polynomials

Laws of Exponents: notes and common mistakes

Study notes

Key ideas

  • An exponent counts how many times the base is a factor.
  • Same base, multiply: add exponents. Same base, divide: subtract exponents.
  • Power of a power: multiply the exponents.
  • Power of a product: apply the power to every factor, including the coefficient.

Steps

  1. Check that the bases match.
  2. Handle coefficients separately (multiply or divide the numbers).
  3. Apply the exponent rule to each variable.
  4. Double-check by writing out the factors.

Formulas

  • Product: am · an = am+n
  • Quotient: am ÷ an = am−n (a ≠ 0)
  • Power of a power: (am)n = amn
  • Power of a product: (ab)n = an bn

Watch out

  • The exponent belongs only to the symbol or parenthesized group directly to its left. In 2x3, only x is cubed. To cube the 2 too, write (2x)3.
  • (2x)3 = 8x3, not 2x3.
  • x3 + x3 = 2x3, not x6. Adding like terms changes the coefficient, not the exponent.

Why it works

  • Why add exponents when multiplying? Each exponent counts x’s, and multiplying merges the two lists of x’s.
  • Why multiply in a power of a power? You are repeating a bundle of x’s several times.

Common mistakes

✗ Mistakex3 · x4 = x12
✓ InsteadAdd the exponents: x3 · x4 = x7. Only a power of a power multiplies exponents.
✗ Mistake(2x)3 = 2x3
✓ InsteadCube both factors: (2x)3 = 8x3.
✗ Mistakex3 + x4 = x7
✓ InsteadDifferent powers are not like terms, so x3 + x4 cannot be combined.

Teach it back

Explain why x3 · x4 = x7 using only the idea of counting x’s.

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

  • I wrote out the x’s: (x · x · x)(x · x · x · x).
  • I counted 7 x’s in a row.
  • I stated the rule: same base, add the exponents.

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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