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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3 Responses
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.
I like to think of an exponent as a “repeat-multiply tag” that sticks to its own base. It only slides around safely through multiplication or division, never through addition or subtraction. That gives two big families of rules. Same base: when you multiply or divide expressions with the same base, you combine exponents (a^m · a^n = a^(m+n), a^m / a^n = a^(m−n)). Same exponent: when you multiply or divide expressions that share the same exponent, you can combine the bases inside one power ((ab)^n = a^n b^n and (a/b)^n = a^n / b^n). There’s no rule for adding different powers-addition is where the “exponent tag” does not move.
Let’s walk your examples. For 3^2 × 5^2, the exponents match, so you can merge the bases: (3×5)^2 = 15^2 = 225. It is not 3^(2+2) (that would be 81), because adding exponents needs the same base. For 10^2 + 10^3, no exponent law applies to addition; you can only factor a common power: 10^2 + 10^3 = 10^2(1 + 10) = 1100, not 10^5. For 8^2 ÷ 4^2, the exponents match, so (8/4)^2 = 2^2 = 4. If you want to use the “subtract exponents” rule, first rewrite with the same base: 8^2/4^2 = (2^3)^2/(2^2)^2 = 2^6/2^4 = 2^(6−4) = 2^2 = 4. What you tried, 8^(2−2), would only be legal if both numerator and denominator had base 8.
A quick memory hook: same base → combine exponents; same exponent → combine bases; plus and minus → hands off the exponents (you can only factor). Khan Academy has a nice summary of these rules if you want a one-page refresher: https://www.khanacademy.org/math/algebra/introduction-to-exponential-functions/properties-of-exponents/a/exponent-properties-in-general. Does that framing help your “hat-tricks” feel more predictable? Want to try a couple more, like 6^3 · 2^3, 9^4 ÷ 3^4, and 5^2 + 5^5, and say which rule (if any) you’d use?
Think of exponents as little hats with two travel rules. Rule A (same base): when the bases match and you’re multiplying or dividing, the hats don’t move; instead, the base teams up by changing how many hats it’s wearing. Multiply same base → add exponents: a^m × a^n = a^(m+n). Divide same base → subtract exponents: a^m ÷ a^n = a^(m−n). Example: 2^3 × 2^4 = 2^(3+4) = 2^7, and 2^5 ÷ 2^2 = 2^(5−2) = 2^3. Rule B (same exponent): when the exponents match, they travel together as a group over multiplication or division of different bases: a^n × b^n = (ab)^n and a^n ÷ b^n = (a/b)^n. Example: 3^2 × 5^2 = (3×5)^2 = 15^2 = 225. That’s why 3^2 × 5^2 is not 3^(2+2); 3^4 = 81, totally different. And for your division: 8^2 ÷ 4^2 = (8/4)^2 = 2^2 = 4; subtracting exponents only works when the bases are the same, which 8 and 4 aren’t.
Here’s the big “do not enter” sign: exponents do not distribute over addition or subtraction. No matter how tempting, a^m + a^n is not a^(m+n). The best you can do is factor a common power. Worked example: 10^2 + 10^3 = 100 + 1000 = 1100, and you can rewrite it as 10^2(1 + 10) = 10^2 × 11, but it is definitely not 10^(2+3). Memory hook: hats slide over times and divide, but they freeze at plus and minus. Quick checklist: multiply same base → add exponents; divide same base → subtract exponents; same exponent across × or ÷ → combine the bases; never combine powers across + or −.