Radicals and rational exponents: How is "30 = 1" a true statement?

[Illvoices]

Hey colleagues I've moved on to a new type of problem and my first issue is exponents
You see there is a part of this solution that doesn't add up to me.

30=1

Why and when in God's plan does this ever equal to one? :evil:
^{You see I thought that the exponents were blank when they are raised up by zero. Or at least they are multiplied by zero to the variable. A common mistake .}

 

[Denis]

30^0 = 1 ; over and out!

 

[lev888]

And it's related to exponents how?
Please post complete statement of the problem or a part of it that at least makes some sense.

 

[JeffM]

Hey colleagues I've moved on to a new type of problem and my first issue is exponents
You see there is a part of this solution that doesn't add up to me.

30=1

Why and when in God's plan does this ever equal to one? :evil:
^{You see I thought that the exponents were blank when they are raised up by zero. Or at least they are multiplied by zero to the variable. A common mistake .}

A useful way to define exponents is this:

\(\displaystyle \text {Given : } n \in \mathbb Z,\ n \ge 0,\text { and } a > 0,\)

\(\displaystyle a^n = 1 \text { if } n = 0, \text { but } a^n = a * a^{(n-1)} \text { if } n > 0.\)

The result is that we get a definition for

\(\displaystyle a^1 = a^0 * a = 1 * a = a\),

a result that we could never get from "we multiply a by itself 1 time," a totally meaningless phrase because multiplication involves at least 2 numbers. Furthermore, it entails that

\(\displaystyle a^2 = a^1 * a = a * a \text { and } a^3 = a^2 * a = a * a * a\),

which is consistent with the definition that you were taught in grade school.

Defining things this way simplifies the laws of exponents and leads naturally to the definition of negative exponents.

 

[Dr.Peterson]

Hey colleagues I've moved on to a new type of problem and my first issue is exponents
You see there is a part of this solution that doesn't add up to me.

30=1

Why and when in God's plan does this ever equal to one? :evil:
^{You see I thought that the exponents were blank when they are raised up by zero. Or at least they are multiplied by zero to the variable. A common mistake .}

Presumably you meant to ask, not why 30 = 1 (which is not true, of course), but why 3^0 = 1, that is, why 3⁰ = 1.

One nice way to think about it is to define powers a^n as starting with 1 and then multiplying by a, n times. For example,

3⁴ = 1*3*3*3*3, where we multiply by 3, 4 times.
​

That makes a lot more sense than talking about multiplying 3 by itself 4 times!

Then

3⁰ = 1, where we multiplied by 3, 0 times.
​

 

[HallsofIvy]

We can, by simply counting positive integer exponents, get the very nice formula \(\displaystyle a^ma^n= a^{m+n}\) (for example, \(\displaystyle a^3a^2= (a\cdot a\cdot a)(a\cdot a)= a\cdot a\cdot a\cdot a\cdot a= a^5\)). We would like that formula to be true even when the exponents are not both positive integers. In particular, if m= 0 we would like to have \(\displaystyle (a^0)(a^n)= a^{0+ n}= a^n\). In order to have that true we define \(\displaystyle a^0\) to be 1.

(We can extend that further. For a positive integer, n, we would like to have \(\displaystyle a^{n+ (-n)}= a^0= 1\) so we must define \(\displaystyle a^{-n}= \frac{1}{n}\).)