Multiply two powers and the exponents add; raise a power to a power and they multiply. These free printable worksheets show a line where the wrong one of those two was used, and ask which answer is correct.
The two laws are the most confusable pair in early algebra, and the reason is that neither is memorable on its own — they are memorable against each other. A sheet that practises only one teaches a child to apply whichever they last saw, which is why both appear here on the same page.
Why this sheet works
The wrong answers are the real confusions rather than arithmetic slips: exponents multiplied when they should be added, subtracted instead of added, the two bases added as well as their powers, the product turned into a sum. Each one is a rule a student has actually reached for.
How this one works
One question from this sheet, worked through a step at a time.
Spot the mistake: (x³)² · x² = x¹². Which answer is right?
(x³)² · x²
(x³)² · x²=x⁶ · x²
(x³)² · x²=x⁶ · x²
(x³)² · x²=x⁸
(x³)² · x²=x⁸
they wrote:x¹²
Do not check by reading their line. Somebody else's working is the most persuasive thing on the page, and rereading it is how you end up agreeing with it — do the step yourself first, then compare.
On your own, and writing the step nobody writes down: (x³)² · x² becomes x⁶ · x².
Tidy that up and it is x⁸.
Now compare. They wrote x¹², and the last power was multiplied in as well as the first two — so the correct answer is x⁸.
Read each line of working and choose the answer that is correct.
1Spot the mistake: (x³)² · x² = x¹². Which answer is right?x¹²x⁶x⁷x⁸Why?
2On Priya's paper: x³ · x³ ÷ x² = x². Which line should replace it?x²x⁴x⁷x⁸Why?
3Ben wrote (x³)⁴ · x² = x⁹. Which is correct?x¹²x²⁴x⁹x¹⁴Why?
4On Ana's paper: x⁴ · x³ ÷ x² = x⁹. Which line should replace it?x⁵x¹⁰x⁹x³Why?
Algebra reasoning · Exponent lawsAnswer key
1. (x³)² · x² = x⁸, not x¹² (the last power was multiplied in as well as the first two)2. x³ · x³ ÷ x² = x⁴, not x² (the second power was taken away and the third one added)3. (x³)⁴ · x² = x¹⁴, not x⁹ (the stacked powers were added where they should be multiplied)4. x⁴ · x³ ÷ x² = x⁵, not x⁹ (the dividing step added its power instead of taking it away)
