代数表达式化简疑问：为何幂乘积不能改写为乘积的幂？

Hey there, let's clear up this confusion once and for all—both of the "unallowed" simplifications you're asking about are actually mathematically valid! It's likely just a textbook choice to stick to a specific form for teaching purposes, not a hard rule.

针对第一个问题：$km \times mn \times nk = k^2m2n^2$ vs $(kmn)^2$ #

• First, recall the power of a product rule: For any real numbers $a, b$ and positive integer $n$, $(ab)^n = a^n b^n$. This rule works in reverse too: $a^n b^n = (ab)^n$.
• Applying that reverse rule here: $k^2m2n^2$ is exactly $k^2 \times m^2 \times n^2$, which can be rewritten as $(k \times m \times n)^2 = (kmn)^2$. These two expressions are completely equivalent—no mathematical reason you can't use either form.

针对第二个问题：$3pq^2r \times 6p^2qr \times 9pqr^2 = 162p^4q4r^4$ vs $162(pqr)^4$ #

• Same logic applies here. Using the reverse power of a product rule, $p^4q4r^4 = (p \times q \times r)^4$. So multiplying by the coefficient 162 gives you $162(pqr)^4$, which is 100% equal to the textbook's given answer.

Why does the textbook say it "can't be further simplified"? #

This is almost certainly a teaching strategy:

• Those examples are probably designed to reinforce the basics of monomial multiplication: multiplying coefficients first, then adding exponents for like bases. Sticking to the expanded exponent form ($k^2m2n^2$, $162p^4q4r^4$) helps lock in that foundational skill before moving on to more advanced shortcuts like reversing the power of a product rule.
• The textbook might introduce that reverse simplification in a later section focused on exponent rules, so they're holding off for now—not saying it's mathematically incorrect.

内容的提问来源于stack exchange，提问作者Connor Leeming