Doubts about Multiplying Monomials.

Question

I want to clarify proof of multiplying monomials.

1) $c * c ^2 = c^3$ sum power because power is shorter form of multiplication

2) $b^2 * b * 2b^6 = 2b^9$ 2b) - Is it commutative property ?

3) $2^a * 2^b = 2^{a+b}$. Why $2^{a+b}$ instead of $4^{a+b}$? Any proofs?

Thanks!

Answer 1

For the first one, you always add the exponents. It can be written as $c \cdot c \cdot c$. Tell me how many $c$'s do you see?

For the second one commutative property plays no role. If you expand all the monomials and multiply them you will find the answer.

The last one can be written as:

$$\underbrace{c \cdot c \cdots c \cdot c}_{a \text{ times }} \cdot \underbrace{c \cdot c \cdots c \cdot c}_{b \text{ times }}$$

How many times do you see $c$? It's easy. $a+b$ times.