Eighth Grade · Math · week 20

Negative and zero exponents

Understand and apply the meaning of zero and negative integer exponents by extending the pattern of shrinking powers.

Look at a shrinking pattern of powers of 2: 2^3=8, 2^2=4, 2^1=2. Each step down divides by the base (2). Keep dividing: 2^0 = 1, and 2^-1 = 1/2, and 2^-2 = 1/4. So the rule is: any nonzero number to the 0 power is 1, and a negative exponent means "take the reciprocal" -- flip it to 1 over the positive-exponent version.

A negative exponent does NOT make the answer negative. It only means "flip to a fraction." 3^-2 is 1/9, never -9 and never -1/9.

📋 For the teacher
Teach it (10–15 min):
  • Build the shrinking-powers-of-2 (or 10) table together, step by step, and let them notice what 2^0 and 2^-1 must be to keep the divide-by-2 pattern going.
  • Say the rule as a flip: "a negative exponent means take the reciprocal of the positive-exponent version" -- it does not mean the answer is negative.
  • The powers-of-10 version of this pattern (10^-1 = 0.1, 10^-2 = 0.01) is exactly what makes small-number scientific notation work next week -- point out that connection now.
  • Name the common trap directly: 3^-2 is 1/9, not -9 and not -1/9.
Talk about it:
  • Why does it make sense that ANY nonzero number to the 0 power equals 1, based on the dividing pattern?
  • Does a negative exponent make the final answer negative? What does it actually do?
  • How does the powers-of-10 pattern connect decimals like 0.01 to exponents?

Mastery looks like: They evaluate zero and negative integer exponents correctly (as 1, and as a reciprocal fraction) and explain why using the dividing pattern.

Common stumbles: Thinking a negative exponent makes the result negative; forgetting the "1 over" part and just changing the sign of the base instead.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

ExponentsRepeated multiplication and how fast powers grow.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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