Would someone be able to please explain to me what the following means - a^-1 = 1/a, a^-m = 1/a^m

 

I've recently gone over the area of scientific notation and this part in particular didn't make sense.

 

thanks.

[math] x^{-1} = \frac {1}{x^{+1}}= \frac 1x [/math] has nothing to do with scientific notation, as far as I know. It's merely a well-known rule for calculations with exponents.

ok, yes sorry, exponents. But what exactly does it mean or imply ?

X^mXⁿ=X^(m+n)

 

For example

m=1, n=-2

 

X¹X^-2=X^(1-2)=X^-1

X/X²=1/X

Would someone be able to please explain to me what the following means - a^-1 = 1/a, a^-m = 1/a^m

 

Well, if you are dealing with a mathematical structure where [math]a^{-1} = \frac{1}{a}[/math] you are implicitly assuming that this structure admits of multiplication and multiplcative identity (here [math]1[/math]). So that [math]1a = a[/math] and [math]aa^{-1} = a \frac{1}{a} = \frac{a}{a} = 1[/math].This is the definition of the multiplicative inverse.

 

But you should be aware that there are structures where [math]a^{-1}a = 0[/math] which implies that [math]a^{-1} = -a[/math]. For these structures the operation in question is called addition, with zero as the additive identity. Maybe you should ignore that point for now, important though it is.

 

So sticking with multiplication, notice that [math]a= a^1[/math] so you simply replace the exponent [math]1[/math] with the exponent [math]m[/math], and it follows that [math]a^{-m} = \frac{1}{a^m}[/math]