Fractional exponents

  • #1
Can someone explain the logic behind this?

For instance if 2 to the 3rd power = 2 x 2 x 2 =8

So 2 to the 3rd power is telling me I have 2 multiplied by itself 3 times.

Now how would I solve for 2 to the 1/3rd power? It is telling me I have 2 multiplied by itself 1/3 times but how do you solve this?

Can you show me how you would work that out?

Thanks.
 

Answers and Replies

  • #2
pwsnafu
Science Advisor
1,080
85
For a positive number, x, and a positive integer, n, we define the notation
[tex]x^n := x \cdot x \cdot \ldots x[/tex]
where there are n number of multiplications. Just as subtraction inverts addition, and division inverts multiplication, we want a notation that inverts exponentiation. We define
[tex]x^{1/n} := \sqrt[n]{x}[/tex]
In words, the notation [itex]x^{1/n}[/itex] means "find the number, y, such that yn = x".

And that's the "logic". We are free to define notation in any way we want. That's it.

As to why we do this, you can prove that [itex](\sqrt[n]{x})^m = (x^{1/n})^m = (x^m)^{1/n} = \sqrt[n]{x^m}[/itex]. So it behaves like "multiplying" the exponents together.
 
  • #3
I still feel a bit lost. Any chance you can provide an example with numbers?
 
  • #4
pwsnafu
Science Advisor
1,080
85
Well, its better to stick to integers, as the calculations are clearer.

The notation 81/3 means find y, such that we solve y3 = 8. Namely the answer is 2. 2 times 2 times 2 is 8. As you noted.
Observe that we have
[tex]2^3 = 8[/tex]
[tex](2^3)^{1/3} = 8^{1/3}[/tex]
[tex]2^{3 \times 1/3} = 8^{1/3}[/tex]
[tex]2 = 8^{1/3}[/tex]
after I write the equation right to left. So we are inverting equations.

Similarly, 10866832384811/4 = 1021 because
1021 times 1021 times 1021 times 1021 = 1086683238481.

Suppose now we wanted 21/2, which is find y such that y2 = 2. It's the square root of 2. We know that 12 = 1 < 2 <4 = 22. So y is strictly between 1 and 2. It turns out that this number is not a fraction either. Its approximately 1.41421356... never terminating or repeating. A mathematician would simply write [itex]\sqrt{2}[/itex] rather than be bothered to calculate what that number is.

Actually calculating these things by hand is not recommended. Either you memorize a lot of tables, learn how to use log tables, or use a calculator. Or learn some algorithms and spend a lot of time.
 
  • #5
I think my real question is how do you find y to the power of 3 = 8 without intuitively knowing it?
 

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