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3 Responses
Short version: a negative exponent only flips the base it’s actually stuck to. No brackets, no flip for the others. So 2^-3 = 1/8, and in 5x^-2 the -2 is on x only, so it’s 5/x^2. The 5 did nothing wrong-don’t flip it unless the exponent applies to it. If the whole thing is in the base, then yes, flip the whole thing: (5x)^-2 = 1/(5x)^2 = 1/(25x^2). Similarly, 5^-1 x^-2 would be 1/(5x^2).
For fractions and grouped stuff: (2/3)^-2 means invert first, then square, so (3/2)^2 = 9/4. In general, (a/b)^-n = (b/a)^n. And (ab^-1)^-1: inside is a/b, so its reciprocal is b/a. Same result by laws: (ab^-1)^-1 = a^-1(b^-1)^-1 = a^-1 b = b/a. Handy mental trick: a negative exponent means “move it to the other side of the fraction bar.” So x^-3 = 1/x^3, and 1/x^-3 = x^3.
Hope this helps!
You’ve got the key idea: a negative exponent means “take the reciprocal,” but only of whatever the exponent is actually attached to. So 2^-3 = 1/8, and 5x^-2 means the -2 is only on x, so it becomes 5/x^2 and the 5 does not flip. If the negative exponent is on a whole grouped base, then everything in that base is affected. For example, with (5x)^-2 I’d think “reciprocal, then square,” so first 1/(5x) and then square the pieces to get 1/(5x^2). For a fraction like (2/3)^-2, I invert first to 3/2 and then square to get 9/4. And for something like (ab^-1)^-1, inside is a/b, so raising to -1 flips it to b/a, which you can also write as a^-1 b. The main habit is to check what the exponent is attached to: if it’s just the x, only x moves; if brackets pull in more factors, the reciprocal step applies to the whole bracket before you apply the positive power.
You’ve got it-the negative exponent only flips the factor it’s attached to, so 5x^-2 = 5/x^2, but (5x)^-2 = 1/(25x^2). With brackets/fractions, treat the whole thing as the base: (2/3)^-2 = (3/2)^2 = 9/4, and (ab^-1)^-1 = (a/b)^-1 = b/a.