Exponent laws

[astral cape]

I was wondering if someone could help me understand the other exponent rules (excluding the product rule, quotient rule, negative exponent rule, and anything raised to the power of zero). I’d love to know how they work and why they make sense, especially from a logical or pattern-based perspective.

[fierce stone]

@astral cape

The power of a product rule follows a similar logic.

[unkempt scarab]

For fractional exponent rule:
if a^m means you multiply a with itself m times, then (a^(1/m)^m) means it take m multiplications of b = a^(1/m) to get back to a, since, by the power of a power rule, b^m = a^(m/m) = a. That means a^(1/m) = mth root of a, by definition of mth root

[fierce stone]

this is for power of a product rule

[fierce stone]

fractional exponent rule (this is what shsgd said, but typeset a little better)

[astral cape]

Sorry but Why does n root a to the power of n, give you a^bn?

[fierce stone]

power of a power rule

we let n root a = a^b

[fierce stone]

there's a small typo here on the third last line. "Power of a product rule" should be "Power of a power rule"

[astral cape]

Ohh

I get it

[stark siren]

We use the other exponent rule:
x^a ÷ x^a = x^(a - a) = x^0 as a-a will be 0
But we also know that:
x^a ÷ x^a = 1, since any nonzero number divided by itself is 1
So we have:
x^0 = 1
Therefore:
x^0 = 1 for all x ≠ 0

☝️ for power of 0 rule\

[fierce stone]

do .close if youre done

anyways yeah what i just told you was what i reasoned back when i was first introduced to the exponents

it's always good to ask for an intuition behind rules that seemingly fall from the sky

[fierce stone]

even better if you try to come up with one yourself first

[astral cape]

Thanks

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