2. Fractional Exponents
Fractional exponents can be used instead of using the radical sign (√). We use fractional exponents because often they are more convenient, and it can make algebraic operations easier to follow.
Fractional Exponent Laws
The n-th root of a number can be written using the power `1/n`, as follows:
Meaning: The n-th root of a when multiplied by itself n times, gives us a.
a1/n × a1/n × a1/n × ... × a1/n = a
[Multiply n times]
The cube root of `8` is `2` (since `2^3=8`).
We can write the cube root of 8 as:
The following 3 numbers are equivalent:
The square root of a number can be written using the radical sign (√) or with exponent 1/2.
The following are equivalent:
The 4-th root of `625` can be written as either:
or equivalently, as
Its value is `5`, since `5^4= 625`.
So we could write:
`625^(1/4) = root(4)625 = 5`
The number under the radical is called the radicand (in Example 3, the number `625` is the radicand).
Order/Index of the radical
The number indicating the root being taken is called the order (or index) of the radical (in Example 3, the order is `4`).
These definitions are here so you know what your textbook is talking about.
Raising the n-th root to the Power m
If we need to raise the n-th root of a number to the power m (say), we can write this as:
This experssion means we need to "take the n-th root of the number a, then raise the result to the power m". With fractional exponents, we would write this as:
Actually, we get the same final answer if we do it in the other order, "raise a to the power m, then find the n-th root of the result". That is,
But the first one is usually easier to do becuase finding the n-th root first gives us a smaller number, which is then easy to raise to the power m.
Question 1: Evaluate `5^(1//2)5^(3//2)`
Question 2: Evaluate `(1000^(1text(/)3))/(400^(-1text(/)2))`