Fractional Exponents: Rules For Multiplying & Dividing

Learning to deal with exponents forms an integral part of any math education, but thankfully the rules for multiplying and dividing them match the rules for non-fractional exponents. The first step to understanding how to deal with fractional exponents is getting a rundown of what exactly they are, and then you can look at the ways you can combine exponents when they're multiplied or divided and they have the same base. In brief, you add the exponents together when multiplying and subtract one from the other when dividing, provided they have the same base.

Fractional exponents provide a compact and useful way of expressing square, cube and higher roots. The denominator on the exponent tells you what root of the "base" number the term represents. In a term like ​_x^a​, you call ​x​ the base and ​a_​ the exponent. So a fractional exponent tells you:

\(x^{1/2} = \sqrt{x}\)

The denominator of two on the exponent tells you that you're taking the square root of ​x​ in this expression. The same basic rule applies to higher roots:

\(x^{1/3} = \sqrt[3]{x}\)

And

\(x^{1/4} = \sqrt[4]{x}\)

This pattern continues. For a concrete example:

\(9^{1/2} = \sqrt{9}=3\)

And

\(8^{1/3} = \sqrt[3]{8}=2\)

Multiply terms with fractional exponents (provided they have the same base) by adding together the exponents. For example:

\(x^{1/3} × x^{1/3} × x^{1/3} = x^{(1/3 + 1/3 + 1/3)}\)
\(= x^1 = x\)

Since ​x​^1/3 means "the cube root of ​x​," it makes perfect sense that this multiplied by itself twice gives the result ​x​. You may also run into examples like ​x​^1/3 × ​x​^1/3, but you deal with these in exactly the same way:

\(x^{1/3} × x^{1/3} = x^{( 1/3 + 1/3)}\)
\(= x^{2/3}\)

The fact that the expression at the end is still a fractional exponent doesn't make a difference to the process. This can be simplified if you note that ​x​^2/3 = (​x​^1/3)² = ∛​_x_​². With an expression like this, it doesn't matter whether you take the root or the power first. This example illustrates how to calculate these:

\(8^{1/3} + 8^{1/3} = 8^{2/3}\)
\(= (\sqrt[3]{8})^2\)

Since the cube root of 8 is easy to work out, tackle this as follows:

\((\sqrt[3]{8})^2 = 2^2 = 4\)

So this means:

\(8^{1/3} + 8^{1/3}= 4\)

You may also encounter products of fractional exponents with different numbers in the denominators of the fractions, and you can add these exponents in the same way you'd add other fractions. For example:

\(\begin{aligned}
x^{1/4} × x^{1/2} &= x^{(1/4 + 1/2)} \
&= x^{(1/4 + 2/4)} \
&= x^{3/4}
\end{aligned}\)

These are all specific expressions of the general rule for multiplying two expressions with exponents:

\(x^a + x^b = x^{(a + b)}\)

Tackle divisions of two numbers with fractional exponents by subtracting the exponent you're dividing (the divisor) by the one you're dividing (the dividend). For example:

\(x^{1/2} ÷ x^{1/2} = x^{(1/2 – 1/2)}\)
\(= x^0 = 1\)

This makes sense, because any number divided by itself equals one, and this agrees with the standard result that any number raised to a power of 0 equals one. The next example uses numbers as bases and different exponents:

\(\begin{aligned}
16^{1/2} ÷ 16^{1/4} &= 16^{(1/2 – 1/4)}\)
\(&= 16^{(2/4 – 1/4)}\)
\(&= 16^{1/4}\)
\(&= 2
\end{aligned}\)

Which you can also see if you note that 16^1/2 = 4 and 16^1/4 = 2.

As with multiplication, you may also end up with fractional exponents that have a number other than one in the numerator, but you deal with these in the same way.

These simply express the general rule for dividing exponents:

\(x^a ÷ x^b = x^{(a – b)}\)

If the bases on the terms are different, there is no easy way to multiply or divide exponents. In these cases, simply calculate the value of the individual terms and then perform the required operation. The only exception is if the exponent is the same, in which case you can multiply or divide them as follows:

\(x^4 × y^4 = (xy)^4 \
x^4 ÷ y^4 = (x ÷ y)^4\)