Lacey and Haley are rewriting expressions in an equivalent, simpler form.

1. Haley simplified $x^3 · x^2$and got $x^5$. Lacey simplified $x^3 + x^2$ and got the same result! However, their teacher told them that only one simplification is correct. Who simplified correctly and how do you know?

   Answer (a):

   Haley is correct. Haley used the product rule with exponents.
   Since each factor has the same base, we simply add the exponents.

2. Haley simplifies $3^5 \cdot 4^5$ and gets the result $12^{10}$, but Lacey is not sure. Is Haley correct? Be sure to justify your answer.

   Hint (b):

   If Haley is correct, then $2^1\cdot3^1 = 6^2$
   Which would imply that $2\cdot3 = 36!$
   Clearly this counterexample provides a hint of why Haley is wrong.