Fractional Exponents and Radicals

What a one-half power must mean

The product rule for exponents says that to multiply two powers of the same base you add the exponents: $a^m \cdot a^n = a^{m+n}$. That rule was proved for whole-number exponents, but insisting it keeps holding for a fractional one is exactly the choice this lesson makes. Try it on a number first, with $m = n = \tfrac{1}{2}$ and $a = 9$:

Whatever $9^{1/2}$ means, it has to square to give $9$. Nothing about that argument used the number $9$ in particular, so the same reasoning holds for any base $a$:

Read that general line the same way. It says the number $a^{1/2}$, multiplied by itself, gives back $a$. A number that squares to $a$ is exactly what a square root of $a$ is. So if the product rule is to keep holding, $a^{1/2}$ must be a square root of $a$, and the next block pins that down to a single value.

The picture behind this is the one you met when square roots were introduced. A square of area $a$ has side length $\sqrt{a}$, because the side multiplied by itself is the area. The fractional exponent just records that same fact, since $a^{1/2}$ is the number you multiply by itself to reach $a$.

The nth root

The same move works for any root. Instead of splitting the exponent into two halves, split $1$ into $n$ equal pieces of size $\tfrac{1}{n}$. The rule you need now is the power-of-a-power rule, $(a^p)^q = a^{pq}$, which lets you raise $a^{1/n}$ to the $n$th power by multiplying the exponents:

So $a^{1/n}$ is a number whose $n$th power is $a$. That number is called the $n$th root of $a$, and it gets its own symbol, an extension of the radical sign you already use for square roots.

In the symbol $\sqrt[n]{a}$, the small number $n$ is the index of the root and the number $a$ underneath is the radicand. A square root is the case $n = 2$, and its index is left unwritten: $\sqrt{a}$ means $\sqrt[2]{a} = a^{1/2}$. A cube root has index $3$, written $\sqrt[3]{a} = a^{1/3}$. You read $\sqrt[n]{a}$ as “the $n$th root of $a$.”

Keep the even-odd rule from above in mind for every root in this lesson: an even index needs a radicand that is zero or positive, and an odd index accepts any real radicand.

Just as knowing the perfect squares makes square roots quick, knowing the small perfect cubes makes cube roots quick:

Reading the bottom row back to the top gives the cube root directly. Since $27$ sits under $3$, you know $\sqrt[3]{27} = 3$.

Raising to a fraction like two-thirds

A fractional exponent whose top is not $1$, such as $a^{2/3}$, combines a root and a power in one symbol. Try it first on a number before asking what the rule is in general.

Try both orders on $8^{2/3}$ to see why root-first is easier. Root-first gives $\left(\sqrt[3]{8}\right)^2 = 2^2 = 4$, working with the small number $8$. Power-first gives $\sqrt[3]{8^2} = \sqrt[3]{64} = 4$, the same answer but by way of the larger $64$. Same result, less arithmetic on the first route.

That pattern holds for every fractional exponent, not just $\tfrac{2}{3}$. Because $\tfrac{m}{n} = m \cdot \tfrac{1}{n} = \tfrac{1}{n} \cdot m$, the power-of-a-power rule lets you read $a^{m/n}$ in two equivalent ways, one for each order you just tried.

Radicals and fractional exponents are the same idea

Everything so far can be read in one sentence: a radical and a fractional exponent are two names for the same number. Collecting the results, for $a \ge 0$ and positive whole numbers $m$ and $n$,

This is a dictionary you can read in either direction. A radical becomes a fractional exponent by putting the index in the denominator, and a fractional exponent becomes a radical by making the denominator the index. Switching to exponent form is often what makes a messy expression obey the familiar exponent laws. Switching to radical form is often what makes a value easy to evaluate.

Simplifying radicals

Roots split across a product exactly the way powers do. The rule is $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$, and the fractional-exponent form shows why in one line.

Division splits the same way, and for the same reason: writing the root as a fractional exponent and using the matching rule for a power of a quotient, $\left(\tfrac{a}{b}\right)^{p} = \tfrac{a^p}{b^p}$,

with $b \neq 0$ and, for an even index, both $a$ and $b$ zero or positive. For instance, $\dfrac{\sqrt{50}}{\sqrt{2}} = \sqrt{\dfrac{50}{2}} = \sqrt{25} = 5$: one root instead of two.

The product rule earns its keep when you simplify a radical, meaning you pull out any factor that is a perfect power for the index. For a square root you hunt for a perfect-square factor; for a cube root, a perfect-cube factor. Split the radicand into that factor times whatever is left, then root the part that comes out cleanly.

Adding and subtracting radicals

Two radicals are like radicals when they have the same index and the same radicand, such as $2\sqrt{3}$ and $5\sqrt{3}$. Like radicals add and subtract the same way like terms do, because the shared radical behaves as a common factor. Pulling it out is just the distributive law from the last lesson:

Unlike radicals do not combine. There is no rule that adds the radicands: $\sqrt{2} + \sqrt{3}$ is not $\sqrt{5}$, because roots split across products, never across sums. Once each term is in simplest form and the radicands still differ, the sum stays as two terms. Sometimes two radicals only look unlike until you simplify them, and then they turn out to match.

Negative fractional exponents

A negative fractional exponent asks nothing new. It just stacks two ideas you already have: the negative sign means take the reciprocal, and the fraction means take a root and a power. One bookkeeping rule keeps the domain from getting ambiguous: always write $\tfrac{m}{n}$ in lowest terms, with no factor shared by $m$ and $n$, so that an exponent has one settled index to check, not several equivalent fractions that could disagree. (Unreduced, $(-8)^{2/6}$ would look like it needs a sixth root, which rejects a negative base; reduced to $(-8)^{1/3}$, the same exponent plainly allows one.) With $m/n$ in lowest terms, the domain is the same $a^{m/n}$ domain from before, with one extra condition: since $a^{m/n}$ becomes a denominator, it also cannot be zero. So $a^{m/n}$ itself must be defined and nonzero: for even $n$ that means $a > 0$ (an even root needs $a \ge 0$, and $0$ cannot be a denominator), while for odd $n$ any nonzero $a$, positive or negative, still works. Then

Handle the two jobs in order. First the minus sign flips the power into its reciprocal, then the fraction is evaluated as a root and a power just as before. A negative exponent moves the power to the denominator; taking a reciprocal never changes the sign of a nonzero number, so $a^{-m/n}$ is positive whenever $a^{m/n}$ is positive, and negative whenever $a^{m/n}$ is negative (which needs an odd $n$ and a negative $a$, as in $(-8)^{-1/3} = \tfrac{1}{\sqrt[3]{-8}} = \tfrac{1}{-2} = -\tfrac{1}{2}$).

Common mistakes