Solve Fractional Exponents: Logic Explained

[goodabouthood]

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.

 

[pwsnafu]

For a positive number, x, and a positive integer, n, we define the notation
$$x^n := x \cdot x \cdot \ldots x$$
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
$$x^{1/n} := \sqrt[n]{x}$$
In words, the notation $x^{1/n}$ 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 $(\sqrt[n]{x})^m = (x^{1/n})^m = (x^m)^{1/n} = \sqrt[n]{x^m}$. So it behaves like "multiplying" the exponents together.

 

[goodabouthood]

I still feel a bit lost. Any chance you can provide an example with numbers?

 

[pwsnafu]

Well, its better to stick to integers, as the calculations are clearer.

The notation 8^1/3 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
$$2^3 = 8$$
$$(2^3)^{1/3} = 8^{1/3}$$
$$2^{3 \times 1/3} = 8^{1/3}$$
$$2 = 8^{1/3}$$
after I write the equation right to left. So we are inverting equations.

Similarly, 1086683238481^1/4 = 1021 because
1021 times 1021 times 1021 times 1021 = 1086683238481.

Suppose now we wanted 2^1/2, 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 $\sqrt{2}$ 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.

 

[goodabouthood]

I think my real question is how do you find y to the power of 3 = 8 without intuitively knowing it?