Common Mistakes

Frequent Errors

1. Adding exponents when bases differ: $2^x\cdot 3^x\neq 5^x$. 2. Confusing $b^{x+k}$ with $(b^x)^k$ or $b^x+b^k$. 3. Applying $(b^m)^n=b^{m+n}$ instead of $b^{mn}$. 4. Writing $b^{-n}=-b^n$ instead of $1/b^n$. 5. Treating $b^{1/k}$ as $b/k$. 6. Ignoring domain restrictions for even roots of negatives. 7. Forgetting $b^{cx}=(b^c)^x$ changes the base, not just the exponent label. 8. Leaving $2^{x+3}$ unfactored when $8\cdot 2^x$ is clearer.

Why These Happen

Students mix up product vs power rules and import linear addition habits onto exponents.

How to Fix Them

• Same-base check before every simplification.
• Template cards: product (add), power (multiply), negative (reciprocal), root (radical).
• Graph one example per rule to anchor symbolic work.

Avoidance Scripts

• "Same base? Add exponents. Power of power? Multiply exponents."
• "$b^{x+k}=b^k\cdot b^x$ — factor, don't distribute over addition incorrectly."
• "Negative exponent means reciprocal."

Quick Self-Check

• Did I use the correct rule?
• Is the base positive where required?
• Does my rewrite match a known graph move?