Understanding Fractional Exponents in Algebra

Square roots are often represented using radical signs, like this \[\sqrt{9}\]. But, there is another way of representing this. You can use a fractional exponent rather than a radical symbol, as they are more convenient to use.  For example,\[\sqrt{9}\] can be written as 9^1/2.

In Mathematics, fractional exponent also known as rational exponent are expressions that are rational numbers rather than integers. It is an alternate representation for expressing powers and roots together. The general form of fraction exponent is 

\[x^{\frac{a}{b}} = \sqrt[b]{x^{a}}\]

In a fractional exponent, the numerator is the power and the denominator is the root. In the above example, ‘a’ and ‘b’ are positive real numbers, and x is a real number, a is the power and b is the root.

Define Fractional Exponent

A fractional exponent is represented as x^p/q where x is a base and p/q is an exponent. This expression is equivalent to the qth root of x raised to the pth power, or \[\sqrt[q]{x^{n}}\]. For example, \[\sqrt[3]{8^{2}}\] can be written as 82/3. In fractional exponent, the exponent is written before the radical symbol, and also if the base is negative, calculating the root is not simple, instead, it requires complex numbers.

Fractional Exponent Laws

1. The n-th root of a number can be written using the power 1/n as follows:

\[a \frac{1}{n} = \sqrt[n]{a}\]

The n-th root of k when multiplied itself by n times, given us k.

\[k\frac{1}{n} \times k\frac{1}{n} \times k\frac{1}{n} \times . . . . . \times k\frac{1}{n} = k\]

Example:

The cube root of 27 is 9 (as 9³ = 27)

The cube root of 9 can also be written as \[9^{1/3}\] or \[\sqrt[3]{9} = 3\]

The following three numbers are equivalent.

\[9^{1/3}\] or \[\sqrt[3]{9} = 3\]

2. If we need to raise the nth root of a number to the power p(say), we can write this as:

\[x^{p/q} = (\sqrt[q]{x})^{p}\]

The above expression means we need to calculate “the n-th root of a number x, then raise the result to the power p.In the fractional exponent form, we can write this as:

(x^1/q)^p

We can also write the above expression in another way, raise x to the power p, then find the nth root of the result that is

(x^p)^1/q

However, the first method is much easier, but calculating the root gives us a smaller number, which can be easily raised to the power p.

How Negative Fractional Exponent Works?

In negative fractional exponents, we first deal with the negative exponent, then apply the fractional exponent rule. For example,

\[x^{-a} = \frac{1}{x^{a}}\]

\[\frac{1}{x^{-a}} = x^{a}\]

Both of the above equations are true when the variable is a positive real number.

How to Multiply Fractional Exponents With the Same Base?

Multiplying the fractional exponent with the same base means adding the exponents together. For example:

\[y^{1/3} \times y^{1/3} \times y^{1/3} = y^{1} = y\]

As \[y^{1/3}\] means the cube root of y. It means when \[y^{1/3}\] is multiplied thrice, the product is y.

Let us consider any other case:

\[y^{1/3} \times y^{1/3} = y^{(1/3 + 1/3)}\]

= \[x^{2/3}\], this can also be written as \[\sqrt[3]{y^{2}}\]

How to Divide Fractional Exponents With the Same Base?

Dividing the fractional exponent with the same base means subtracting the exponents together. For example:

\[y^{1/3} \div y^{1/3} = x^{0} = 1\]

This implies that any number,  when divided by itself,  is equivalent to 1, and the zero exponent rule says that any number raised to an exponent of 0 is equal to 1.

Here are some examples that show how radical expressions can be rewritten using fractional exponents.

Rewriting Radicals Expressions Using Fractional Exponents

Rewriting Radicals in Fractional Exponent Form With Numerators Other Than One

Solved Examples

1. Calculate \[(8 x^{2} y^{4})^{1/3}\]

Solution:

\[(8)^{1/3} (x^{2}) (y^{4})^{1/3}\]

\[= 2 x^{2/3} y^{4/3}\]

Here, we used the rule:

\[(a^{m})^{n} = a^{mn}\]

We took each item from the bracket and raised to power ⅓. we have done because each item in the bracket is multiplied (if they were added or subtracted, multiplying the items won't be possible).

2. Rewrite \[\sqrt[5]{x^{8}}\] using a fractional exponent.

Solution:

Using the definition of \[a^{m/n}\], we get,

\[\sqrt[5]{x^{8}} = x^{8/5}\]

3. Rewrite \[\sqrt{39}\] using a fractional exponents

Solution:

Using the definition of \[a^{1/n}\], we get:

\[\sqrt{39} = 39^{1/2}\].

4. Rewrite \[y^{3/7}\] using radicals

Solution:

Using the definition of \[a^{m/n}\], we get,

\[y^{3/7} = \sqrt[7]{x^{3}}\]