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:

`a^(1/n)=root(n)a`

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]

Example 1

The cube root of `8` is `2` (since `2^3=8`).

We can write the cube root of 8 as:

`8^(1//3)`

or

`root(\sf 3)8`

The following 3 numbers are equivalent:

`8^(\sf{1/3})=root(\sf 3)8=2`

Example 2

The square root of a number can be written using the radical sign (√) or with exponent 1/2.

The following are equivalent:

`sqrt(100)=100^(1/2)=10`

Example 3

The 4-th root of `625` can be written as either:

6251/4

or equivalently, as

`root(4)625`

Its value is `5`, since `5^4= 625`.

So we could write:

`625^(1/4) = root(4)625 = 5`

Definitions

Radicand

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:

`a^(m/n)=(root(n)a)^m`

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:

`(a^(1//n))^m`

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,

`(a^m)^(1//n)`

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.

Example 4

Evaluate `8^(2/3)`

Example 5

Simplify `(8a^2b^4)^(1/3)`

Example 6

Simplify `a^(3text(/)4)a^(4text(/)5)`

Example 7

Simplify

`((4^(-3/2)x^(2/3)y^(-7/4))/(2^(3/2)x^(-1/3)y^(3/4)))^(2/3)`

Exercises

Question 1: Evaluate `5^(1//2)5^(3//2)`

Question 2: Evaluate `(1000^(1text(/)3))/(400^(-1text(/)2))`

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