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Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k\;$?

I'm aware that $3 = 3^1$ but I would expect $3\cdot 3^k\;$ to be $\;9^k$ or $\;9^{k+1}$.

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Try $k=2$ and see whether it's enlightening. –  Daan Michiels Nov 30 '12 at 16:34
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What's the difference between $3\cdot 3^k$ and $(3\cdot 3)^k$? –  Neal Nov 30 '12 at 16:35
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General fact about mathematics: Nothing Is True, I repeat, Nothing Is True. Unless there’s a reason for it to be true. If you see an appealing-looking formula, your impulse should be to say, “that’s not true”, and then try some numbers to see whether perchance they fit the proposed formula. In particular, $3\cdot3^2=3\cdot3\cdot3=27$, in the case $k=2$, but $9^2=9\cdot9=3\cdot3\cdot3\cdot3=81$. Appealing or no, the formula is false. –  Lubin Nov 30 '12 at 19:57
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5 Answers

up vote 16 down vote accepted

$$3\cdot 3^k = 3\cdot\underbrace{3\cdot 3 \cdot \cdots 3 \cdot 3}_{k\;factors \;of \;3}$$ $$= \underbrace{3\cdot 3 \cdot \cdots 3 \cdot 3}_{k+1\;factors \;of \;3}$$ $$= 3^{k+1}$$

By convention, taking powers (exponentiation) precedes multiplication, and so it is performed first (before multiplication). That is, think of $\;3\cdot 3^k\;$ as expressing $\;3(3^k) = 3^{k+1}$.

If multiplication is to precede exponentiation, one would need to use parentheses to indicate "multiply first, then take the exponent of that product": $(3\cdot 3)^k = 9^k$.

$$\text{But}\;\; 3\cdot (3^k) \neq (3\cdot 3)^k$$ $$3^{k+1} \neq 9^k \text{ unless k = 1}. $$


If you really like the number $9$ and really want to use it in expressing $3\cdot3^k$, then note that $$3\cdot 3^k = 3\cdot\underbrace{3\cdot 3 \cdot \cdots 3 \cdot 3}_{k \;factors \;of \;3} = 9\cdot 3^{k-1} = 3\cdot 3 \cdot \underbrace{3\cdot 3 \cdot \cdots 3 \cdot 3}_{(k-1) \;factors \;of \;3}$$

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The answer is as follows: Because there's a convention that powers have larger priority than multiplication, therefore $$ 3\cdot 3^k = 3\cdot(3^k) \qquad\text{ and not }\qquad 3\cdot 3 ^k = (3\cdot 3)^k.$$

The question is similar to asking why $3+4\cdot 5=3+20=23$ and not $3+4\cdot5=7\cdot5=35$.

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From my point of view, The Answer is: It just does not follow law of indices, and nothing more

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We have $a^b\cdot a^c=a^{b+c}$. Look at this: $$\underbrace{a\cdot a\cdots \cdot a}_{b\text{ factors}}\cdot \underbrace{a\cdot a\cdots \cdot a}_{c\text{ factors}}=\underbrace{a\cdot a\cdots \cdot a}_{b+c\text{ factors}}$$

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By convention $\rm\: a\cdot b^{\,n}\:$ means $\rm\:a \cdot (b^n)\:$ not $\rm\:(a\cdot b)^n.\:$ If we adopted the alternative convention, then many common arithmetical expressions (e.g. polynomials) would require more symbols to notate.

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