为何$(3a) \times(3b)$与$3a3b$等于$9ab$而非$3ab$？

Hey there! Let’s work through your two algebra questions clearly—we’ll stick to fundamental rules, not arbitrary "tricks."

1. Why does $(3a) \times (3b) = 9ab$ instead of $3ab$? #

This boils down to the commutative and associative properties of multiplication, core math rules, not tricks. Here’s the breakdown:

• $(3a)$ is shorthand for $3 \times a$, and $(3b)$ means $3 \times b$.
• When multiplying these together, you’re really calculating $(3 \times a) \times (3 \times b)$.
• Multiplication lets us rearrange and group terms freely, so we can rewrite this as $(3 \times 3) \times (a \times b)$.
• $3 \times 3 = 9$, and $a \times b = ab$, so that gives us $9ab$.

To test with real numbers, plug in $a=2$, $b=4$:

• $(3a) \times (3b) = (6) \times (12) = 72$
• $9ab = 9 \times 2 \times 4 = 72$
• $3ab = 3 \times 2 \times 4 = 24$ (which doesn’t match the actual product, so it can’t be right)

2. Is treating $3a3b$ as $33a*b$ just a "trick"? #

Nope—this is a standard algebra writing rule, not a trick. Let’s first fix your mistake in the $(a+a+a) \cdot (b+b+b)$ derivation:
When expanding $(a+a+a)(b+b+b)$, you need to multiply every term in the first parentheses by every term in the second (the distributive property):
$$
\begin{align*}
(a+a+a)(b+b+b) &= a \cdot b + a \cdot b + a \cdot b + a \cdot b + a \cdot b + a \cdot b + a \cdot b + a \cdot b + a \cdot b \
&= 9ab
\end{align*}
$$
You probably only added three $ab$ terms instead of nine—easy mistake! Each of the three $a$s multiplies each of the three $b$s, giving 9 total products of $ab$.

Now, about $3a3b$: In algebra, we have unwritten shorthand rules:

• A number next to a letter means multiplication (e.g., $3a = 3 \times a$)
• When two such "terms" are written next to each other without an operator, it means multiplication (e.g., $ab = a \times b$, $3a \times 3b$ can be shortened to $3a3b$)

So $3a3b$ is just a cleaner way to write $(3 \times a) \times (3 \times b)$, which simplifies to $9ab$ using the same commutative/associative rules as before. It’s not a trick—it’s just how we write multiplication in algebra to keep things concise.

Quick confirmation test: #

If $a=1$, $b=1$, then $3a3b = 3(1)3(1) = 3 \times 3 = 9$, and $9ab = 9(1)(1) = 9$. If it were $3ab$, that would be 3, which doesn’t match. So $9ab$ is the only correct result.

内容的提问来源于stack exchange，提问作者flabber wabber bit