I’m revising my fundamentals on index notation, and I keep tripping over when I should add, subtract, or multiply the exponents. I feel like I know the rules in isolation, but as soon as they’re mixed together, my brain does a little cartwheel.
For example, how would you simplify (a^3 * a^-5) / a^2? My attempt was: multiply same base → add exponents, so a^3 * a^-5 becomes a^-2. Then dividing by a^2 made me subtract again, so I wrote a^-4. But then I freeze because I’m not sure if that’s okay as-is or if I’m supposed to rewrite it in another form, and I get tangled with the negative sign and the numerator/denominator idea.
Another thing: powers on powers versus chained exponents. I think (2^3)^2 means “power of a power,” so I want to multiply the exponents there. But for 2^3^2, I’m not sure how the order works – is that 2^(3^2) or (2^3)^2? My gut wants to say both are just 2^6 because 3×2=6, but I’m pretty sure that’s me over-simplifying.
And then zero and fractional indices throw me off. Is x^0 really 1 for any nonzero x? It feels like a magic trick I don’t fully get. Also, x^(1/2) – is that the same as √x, or am I mixing that up with 1/(x^2)? When negatives get involved (like x^-1/2), I start second-guessing whether that’s 1/√x or something like √(1/x) and whether parentheses change that meaning.
I’m trying to strengthen my basics so I stop making the same mistakes. If anyone can show a clean way to think about these (maybe a small checklist for when to add, subtract, or multiply exponents, and how parentheses change things), that would be awesome. Any help appreciated!
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3 Responses
A tidy way to think about exponents is: they count net copies of the base. Positive exponents count factors in the numerator; negative exponents count factors in the denominator. This lets you add and subtract exponents like ordinary integers when you multiply or divide same bases.
Your example
(a^3 · a^-5) / a^2
Combine same base a across the whole expression by adding exponents algebraically:
3 + (−5) − 2 = −4
So the result is a^-4. By definition of negative powers, a^-4 = 1/a^4 (as long as a ≠ 0). Both forms are correct; 1/a^4 is often clearer.
Why the basic rules are what they are
– Product rule (same base): a^m · a^n = a^(m+n). You are just pooling factors.
– Quotient rule (same base): a^m / a^n = a^(m−n). Factors in the denominator subtract.
– Power of a power: (a^m)^n = a^(mn). You have m copies, n times.
– Negative exponents: a^(−n) = 1/a^n. This extends the quotient rule to n > m.
– Zero exponent: a^0 = 1 for a ≠ 0, because a^n / a^n = a^(n−n) = a^0 must equal 1.
– Fractional exponents (real numbers, a > 0): a^(p/q) = (qth root of a)^p = qth root of (a^p).
Parentheses and order
– Exponents bind tightly. 1/x^2 means 1/(x^2), not (1/x)^2. Write x^(−1/2), not x^−1/2, to avoid ambiguity.
– Power of a power multiplies exponents only when it is actually a power of a power: (2^3)^2 = 2^(3·2) = 2^6 = 64.
– Chained exponents are evaluated top-down (right-associative) in standard mathematics: 2^3^2 means 2^(3^2) = 2^9 = 512, not (2^3)^2.
– A minus sign is not part of the base unless grouped: −2^2 = −(2^2) = −4, but (−2)^2 = 4.
– Exponents distribute over products and quotients: (ab)^n = a^n b^n, (a/b)^n = a^n/b^n. They do not distribute over sums: (a + b)^n ≠ a^n + b^n in general.
Interpreting fractional and negative exponents
– x^0 = 1 for x ≠ 0.
– x^(1/2) is √x (in reals, x ≥ 0).
– x^(−1/2) = 1/x^(1/2) = 1/√x. Equivalently √(1/x) when x > 0.
– More generally, x^(p/q) = (√[q]{x})^p. Example: 27^(2/3) = (√[3]{27})^2 = 3^2 = 9.
– Be careful with negative bases and even roots in the reals: (−8)^(2/3) = ((−8)^(1/3))^2 = (−2)^2 = 4 is fine, but (−8)^(1/2) is not a real number.
A compact checklist
Add exponents:
– Multiply same base: a^m · a^n = a^(m+n).
Subtract exponents:
– Divide same base: a^m / a^n = a^(m−n).
Multiply exponents:
– Power of a power: (a^m)^n = a^(mn).
Move factors across the fraction bar:
– Negative exponent flips side: a^(−n) = 1/a^n, 1/a^(−n) = a^n.
Roots and fractions:
– a^(p/q) = qth root of a^p (a > 0 for real work).
Parentheses and order:
– Exponent applies only to its base. Use parentheses to show the base.
– 2^3^2 = 2^(3^2) (right-associative). Only (2^3)^2 equals 2^(3·2).
Two simple worked examples
1) Simplify (x^−1 y^3)^2 / (x^2 y^−1).
– Numerator: (x^−1)^2 (y^3)^2 = x^−2 y^6.
– Divide by x^2 y^−1: x^(−2 − 2) y^(6 − (−1)) = x^−4 y^7 = y^7 / x^4.
2) Evaluate 16^(−3/2).
– 16^(−3/2) = 1 / 16^(3/2) = 1 / ( (√16)^3 ) = 1 / 64.
Common pitfalls to avoid
– You cannot add exponents when bases differ: a^m b^m = (ab)^m, but a^m + b^m does not simplify.
– Do not turn a^(m+n) into a^m + a^n or a^(mn) unless there is a power-of-a-power structure.
– Remember 1/x^2 = 1/(x^2), and x^(−1/2) means 1/√x, not √x/1 or something else.
If you keep “count net copies of the base” in mind and let parentheses show the base clearly, the rules fall into place.
You’re on the right track-those “cartwheels” are mostly about keeping straight which rule you’re using and what the parentheses are grabbing. For your example: (a^3 · a^−5)/a^2 = a^(3−5)/a^2 = a^−2/a^2 = a^(−2−2) = a^−4, which is 1/a^4 (for a ≠ 0). Negative exponents just mean “reciprocal,” so a^−4 lives more cleanly as 1/a^4. Power of a power: (2^3)^2 = 2^(3·2) = 2^6. But chained exponents associate to the right: 2^3^2 means 2^(3^2) = 2^9, not (2^3)^2-parentheses are everything here. Zero exponents: x^0 = 1 for x ≠ 0 because a^m/a^m = a^(m−m) = a^0 must equal 1. Fractional exponents: x^(1/2) is √x (for x ≥ 0), and in general x^(p/q) = the qth root of x^p. Combine that with negatives: x^(−1/2) = 1/x^(1/2) = 1/√x (and for positive x, that equals √(1/x)). A tiny checklist I use (and still peek at when I’m sleepy): same base multiply → add exponents; same base divide → subtract; power of a power → multiply; negative exponent → flip to the other side of the fraction bar; fractional exponent → roots; no parentheses in a^b^c → read it as a^(b^c). I still double-check parentheses because different notations can look sneaky, but these rules won’t steer you wrong.
I like to think of exponents as a “tally of copies,” and the rules are just bookkeeping: when you combine the same base, you add tallies; when you split, you subtract; when you take a power of a power, you’re repeating the same packaging, so you multiply. For your example, (a^3 · a^-5)/a^2 = a^(3 + (-5) – 2) = a^-4, which is perfectly fine, and usually rewritten without negative exponents as 1/a^4 (with a ≠ 0). Negative exponents mean “put it in the denominator” (a^-k = 1/a^k), and zero means “everything canceled”: x^m/x^m = x^(m-m) = x^0, but the left side is 1, so x^0 = 1 for any nonzero x. Fractional exponents are roots: x^(1/2) = √x (over the reals, x ≥ 0), and in general x^(p/q) = (qth root of x)^p; adding a negative just flips it: x^(-1/2) = 1/√x, which is the same as √(1/x) when x > 0. Parentheses are the traffic cones that control who the exponent sticks to: (2^3)^2 is a power of a power, so 2^(3·2) = 2^6 = 64, but 2^3^2 is interpreted as 2^(3^2) = 2^9 = 512 (exponentiation associates to the right unless parentheses say otherwise). Quick mental checklist: multiply same base → add exponents; divide → subtract; power of a power → multiply; negative → reciprocal; zero → 1; fractional → roots; and always check the parentheses to see what the exponent actually “covers.” Want to try a couple together, like simplifying (x^-3 y^4)^(-1/2) or deciding the value of 3^-2^2 versus (3^-2)^2?