咨询指数运算变形的代数原理：$8(8^k)-3(3^k)$到$(8^k-3^k)8+5(3^k)$

Hey there! Let's break down this algebraic transformation step by step—it's all about basic distributive property and a simple term-splitting trick, so it's totally manageable once you see the logic.

The Core Idea: Term Splitting & Distributive Property #

The key here is to rewrite one of the terms in the original expression to match the structure of the target form, then factor out a common term. Let's walk through it:

1. Start with the original expression:
   $$8(8^k) - 3(3^k)$$

2. Notice that the target form has a term $(8^k - 3^k)8$. If we expand this, we get $8(8^k) - 8(3^k)$. Compare this to our original expression—we have $-3(3^k)$ instead of $-8(3^k)$. So we can split the $-3(3^k)$ into two parts to bridge this gap:
   $$-3(3^k) = -8(3^k) + 5(3^k)$$

3. Substitute this split back into the original expression:
   $$8(8^k) - 8(3^k) + 5(3^k)$$

4. Now factor out the common factor of 8 from the first two terms:
   $$8(8^k - 3^k) + 5(3^k)$$

That's exactly the transformed form you were curious about!

Quick Verification with Example Values #

Let's test with $k=1$ to confirm:

• Original expression: $8(8^1) - 3(3^1) = 64 - 9 = 55$
• Transformed expression: $(8^1 - 3^1)8 + 5(3^1) = 5*8 + 15 = 40 + 15 = 55$

Another test with $k=2$:

• Original expression: $8(8^2) - 3(3^2) = 864 - 39 = 512 - 27 = 485$
• Transformed expression: $(8^2 - 3^2)8 + 5(3^2) = 55*8 + 45 = 440 + 45 = 485$

Perfect, both forms give the same result every time.

内容的提问来源于stack exchange，提问作者eewf ewfwf