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(3)8

The following 3 numbers are equivalent:

8^{1/3}=root(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

Continues below

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

The number under the radical is called the radicand (in Example 3, the number 625 is the radicand).

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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