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: $a^m \cdot a^n = a^{m+n}$. Nothing in that rule cares whether the exponents are whole numbers, so take it at its word with $m = n = \tfrac{1}{2}$:

Read that line slowly. It says the number $a^{1/2}$, multiplied by itself, gives back $a$. A number that squares to $a$ is exactly what the square root of $a$ is. So if the product rule is to keep holding, $a^{1/2}$ has no room to be anything else: it must be $\sqrt{a}$.

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 $a$.”

These roots stay in the real numbers as long as you mind the index. When $n$ is even, the radicand must be zero or positive, and $\sqrt[n]{a}$ names the non-negative (principal) root, exactly as a square root does. That restriction holds because no real number raised to an even power comes out negative. When $n$ is odd, a negative radicand is perfectly fine and gives a single negative root, so $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$. Every root in this lesson respects that rule, so it always denotes a real number.

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. To see how, split the fraction into a product. 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.

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.

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,

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 the same care for even indices that the radicands be zero or positive. So a quotient of roots is the root of the quotient. That is exactly what lets you divide two radicals of the same index under a single radical sign.

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 way to merge $\sqrt{2} + \sqrt{3}$ into a single radical, and in particular it is not $\sqrt{5}$, because roots split across products, never across sums. 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. So

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. As always, a negative exponent moves the power to the denominator; it does not touch the sign of the result.

Common mistakes