HighSchoolQ

➕ Algebra 1 · Exponents and Polynomials

Zero and Negative Exponents: notes and common mistakes

Study notes

Key ideas

  • x0 = 1 for every nonzero x. A zero exponent does not give 0. In this course 00 is left undefined.
  • x−n = 1/xn for x ≠ 0. A negative exponent means a reciprocal, not a negative number.
  • Both meanings are chosen to keep the quotient rule xm ÷ xn = xm−n working even when the exponent on top is not bigger.
  • The product, quotient, power-of-a-power and power-of-a-product rules still work with zero and negative exponents.
  • Unless told otherwise, give final answers with positive exponents.

Steps

  1. Apply the power rules to any parentheses first (power of a product, power of a power).
  2. Combine matching bases: add exponents when multiplying, subtract when dividing. Treat coefficients as ordinary numbers.
  3. Move each factor with a negative exponent across the fraction bar and make its exponent positive.
  4. Check by substituting a small number, such as x = 2, into the original and into your answer.

Formulas

  • Zero exponent: a0 = 1 (a ≠ 0)
  • Negative exponent: a−n = 1/an (a ≠ 0)
  • Moving back up: 1/a−n = an (a ≠ 0)
  • Fraction base: (a/b)−n = (b/a)n (a ≠ 0, b ≠ 0)

Watch out

  • A negative exponent does not make the number negative: 2−3 = 1/8, not −8. A power b−n is negative only when the base b is negative and the exponent is odd, for example (−2)−3 = −1/8.
  • The exponent belongs only to what is directly to its left: 3x−2 = 3/x2, but (3x)−2 = 1/(9x2).
  • A zero exponent on a letter does not touch the coefficient: 5x0 = 5, but (5x)0 = 1 (with x ≠ 0). Also −40 = −1, but (−4)0 = 1.
  • Negative exponents move factors, not terms that are added: (x + y)−1 = 1/(x + y), not 1/x + 1/y.
  • A negative exponent does not always shrink a number. It flips the base. 2−3 = 1/8 is small, but (1/2)−3 = 23 = 8 is big.

Why it works

  • Why is x0 = 1? Take x3 ÷ x3. It equals 1 (a nonzero number divided by itself), and the quotient rule would give x3−3 = x0. We define x0 = 1 so that the quotient rule keeps working.
  • Why is x−n = 1/xn? Take x1 ÷ x4. Cancelling gives 1/x3, and the quotient rule would give x1−4 = x−3. We define x−3 = 1/x3 so that the quotient rule keeps working.
  • Why x ≠ 0? If x = 0, then x3 ÷ x3 would be 0 ÷ 0, which is undefined, so the argument fails.

Common mistakes

✗ Mistake2−3 = −8 (or −6). A negative exponent makes the number negative.
✓ InsteadA negative exponent means reciprocal: 2−3 = 1/23 = 1/8, which is positive.
✗ Mistakex0 = 0.
✓ Insteadx0 = 1 for any nonzero x.
✗ Mistake3x−2 = 1/(3x2).
✓ InsteadOnly x has the −2: 3x−2 = 3/x2. For the 3 to flip too, you would need (3x)−2, which equals 1/(9x2).
✗ Mistake−40 = 1.
✓ Instead−40 = −(40) = −1. Only (−4)0 equals 1.
✗ Mistake(x + y)−1 = 1/x + 1/y.
✓ Instead(x + y)−1 = 1/(x + y). A negative exponent flips the whole parenthesized group, and a power does not split over addition.

Teach it back

Explain to a friend why x0 = 1 and why x−2 = 1/x2, using the idea of cancelling x’s and the quotient rule.

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

  • I wrote x4 ÷ x4, said it equals 1 and that the quotient rule would give x4−4 = x0, so x0 must mean 1 for the rule to keep working.
  • I wrote x1 ÷ x3 = x/(x · x · x), cancelled one x, and got 1/x2. I said the quotient rule would give x1−3 = x−2, so we define x−2 = 1/x2.
  • I said that a negative exponent does not make the answer negative. It flips the factor to the other side of the fraction bar.

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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