I’m trying to befriend index notation (aka powers), but the little superscript hats keep swapping places when I’m not looking. I think I’ve invented a bogus rule and now I can’t unsee it.
Here’s where my brain does a somersault: when I see something like 3^2 × 5^2, I instinctively try to smoosh it into 3^(2+2). But then another part of me whispers, “Wait, isn’t it also (3×5)^2?” and those two ideas don’t match. So… which universe is real?
Some context for my powers saga:
– I’m pretty sure about this one: 2^3 × 2^4 = 2^(3+4). That feels fine because the base is the same.
– But then for 10^2 + 10^3, my mischievous brain tries to make it 10^(2+3). Yet 100 + 1000 definitely doesn’t look like 10^5, so clearly that shortcut is garbage.
– And with division, I did 8^2 ÷ 4^2 and tried the “subtract the exponents” thing on the 8, getting 8^(2−2) = 8^0, which seems super suspicious. Another voice says maybe it should be (8/4)^2 instead?
I think my incorrect assumption is: “If anything shares a 2 in the air, I can drag the 2 around and do whatever I want.” That feels… illegal.
Could someone help me untangle which rules go with:
– same base vs. same exponent,
– multiplying vs. adding expressions with powers,
– and how to spot when I’m about to do a forbidden exponent move?
If it helps, a simple example walk-through with 3^2 × 5^2 and 10^2 + 10^3 would probably reveal where my hat-tricks are going wrong. I’d love a memory-friendly way to keep these rules straight!
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One Response
The index rules split cleanly into two families, and they only apply to multiplication or division, not to addition. Same base: a^m × a^n = a^(m+n) and a^m ÷ a^n = a^(m−n) (a ≠ 0), so 2^3 × 2^4 = 2^7 is fine, but you cannot use this when the bases differ. Same exponent: a^k × b^k = (ab)^k and a^k ÷ b^k = (a/b)^k, so 3^2 × 5^2 = (3×5)^2 = 15^2, and 8^2 ÷ 4^2 = (8/4)^2 = 2^2 = 4. Notice 3^2 × 5^2 is not 3^(2+2), because adding exponents needs the same base. With addition there is no combining rule: 10^2 + 10^3 ≠ 10^5; the best you can do is factor a common power, 10^2 + 10^3 = 10^2(1 + 10) = 1100. A small analogy: think of exponents as hats stuck to their own base-hats can stack on the same person when you multiply the same base, or a group wearing the same hat can stand under one big hat when you multiply or divide them, but hats never jump across a plus sign. To spot illegal moves, watch for (i) trying to move an exponent across + or −, (ii) adding or subtracting exponents when the bases aren’t the same, and (iii) mixing the two families. Quick checklist: multiply same base → add exponents; divide same base → subtract exponents; multiply/divide same exponent → combine bases and keep the exponent; addition/subtraction → only factor, never fuse.