I’m struggling with the index laws, especially how negative and zero exponents behave and when I’m allowed to combine powers. I keep second-guessing myself about the product rule, quotient rule, and “power of a power,” and I think I’m mixing up signs.
Could someone explain, step by step, why these are (or aren’t) correct, and what the right way to think about them is?
– Is 2^3 * 2^-5 the same as 2^(3-5), or should I interpret that another way?
– Why is x^0 = 1 for a nonzero x? What exactly happens at x = 0?
– Are (3^2)^-1 and 3^(2 * -1) intended to be equal?
– For division, is a^5 / a^-2 equal to a^(5 – (-2)) or a^(5 + (-2))? I get lost with the minus signs.
– Does (ab)^n always become a^n b^n, and does the reverse always hold? What about (a + b)^n?
– With a negative exponent on a product, e.g., (2x^-3 y)^-2, what’s the cleanest way to rewrite it?
– Simple number check: is 4^0 * 4^-2 just 4^-2, and what does that mean as an actual number?
If you can also point out common traps (like where parentheses matter, e.g., -2^2 vs (-2)^2) and a reliable mental checklist to avoid sign mistakes, that would really help me understand the reasoning, not just memorize rules.
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3 Responses
Think of exponents as tidy bookkeeping for repeated multiplication: when the base matches, multiply means “add the exponents,” divide means “subtract (top minus bottom),” a power of a power means “multiply the exponents,” a negative exponent means “flip to the other side of the fraction,” and a zero exponent gives 1 as long as the base isn’t zero. So 2^3 * 2^-5 = 2^(3 + (-5)) = 2^-2 = 1/4; x^0 = 1 for x ≠ 0 because a^m / a^m = a^(m−m) = a^0 must be 1, while 0^0 is left undefined in basic algebra; (3^2)^-1 = 3^(2·-1) = 3^-2 = 1/9; a^5 / a^-2 = a^(5 − (−2)) = a^7 (since subtracting a negative adds); (ab)^n = a^n b^n for integer n (and the reverse direction is the same identity), but (a + b)^n does not split that way-cross-terms appear unless n = 1; with (2x^-3 y)^-2, distribute the -2: 2^-2 x^6 y^-2 = x^6 / (4y^2); and 4^0 * 4^-2 = 4^-2 = 1/16 as a number. Parentheses really matter: -2^2 is -(2^2) = -4, while (-2)^2 = 4; always wrap negatives you want included in the power. My personal breakthrough was a neon sticky note on my desk-“same base add, divide subtract, power-power multiply, negative means flip”-after I lost points on that -2^2 vs (-2)^2 trap; it felt like labeling the spice jars so I wouldn’t grab sugar instead of salt. If you want a clear walkthrough with practice, this Khan Academy page is great: https://www.khanacademy.org/math/algebra/exponents-radicals.
For a common base a ≠ 0 the index laws are: a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), and a^(−k) = 1/a^k. The zero exponent is set so that a^m / a^m = a^(m−m) = a^0 equals 1, which only makes sense when a ≠ 0; at a = 0 we have 0^n = 0 for n > 0, but 0^0 and 0 to a negative exponent are undefined. Using these: 2^3 · 2^(−5) = 2^(3+ (−5)) = 2^(−2) = 1/4. Also (3^2)^(−1) = 3^(2·(−1)) = 3^(−2) = 1/9. For division, a^5 / a^(−2) = a^(5 − (−2)) = a^7; the two minus signs add.
Products distribute cleanly: for any integer n, (ab)^n = a^n b^n, and conversely a^n b^n = (ab)^n (with a,b ≠ 0 if n is negative). This does not work for sums: in general (a + b)^n ≠ a^n + b^n. With a negative exponent on a product, simplify factor by factor: (2 x^(−3) y)^(−2) = 2^(−2) x^6 y^(−2) = x^6/(4 y^2). Number check: 4^0 · 4^(−2) = 1 · 4^(−2) = 1/4^2 = 1/16. Parentheses matter: −2^2 means −(2^2) = −4, while (−2)^2 = 4 (and (−2)^3 = −8). A quick checklist: same base multiply → add exponents; divide → subtract exponents; power of a power → multiply exponents; negative exponent → reciprocal; zero exponent → 1 (nonzero base); distribute over products and quotients, not over sums; use parentheses to control signs.
For same base multiply, add exponents, so 2^3·2^-5 = 2^(3+(-5)) = 2^-2 = 1/4; x^0 = 1 for x ≠ 0 because a^m/a^m = a^(m-m) = a^0 = 1, while 0^0 is undefined; (3^2)^-1 = 3^(2·(-1)) = 3^-2 = 1/9; a^5/a^-2 = a^(5-(-2)) = a^7; (ab)^n = a^n b^n and a^n b^n = (ab)^n, but (a + b)^n ≠ a^n + b^n; (2x^-3 y)^-2 = 2^-2 x^6 y^-2 = x^6/(4y^2); and 4^0·4^-2 = 4^-2 = 1/16.
Checklist: negative exponent → reciprocal, zero exponent → 1 (base ≠ 0), multiply → add exponents, divide → subtract, power of a power → multiply, exponents distribute over products/quotients but not sums, and use parentheses so -2^2 = -(2^2) = -4 while (-2)^2 = 4.