What is a radical? What is a root?

What is a radical? What is a root?

You can think about radicals (also called “roots”) as the opposite of exponents.

We already know that the expression $x^2$ with the exponent of $2$ means “multiply $x$ by itself two times”. The opposite operation would be “what do we have to multiply by itself two times in order to get $x^2$?”

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That’s where radicals come in. If you see

$\sqrt{x}$

it means “the number you have to multiply by itself to get $x$.” The symbol that contains the $x$ is called the “radical sign,” and the expression inside the symbol - in this case $x$ - is called the “radicand.” Instead of saying that an expression is inside the radical sign, however, we usually say that it’s under the radical sign. When you see $\sqrt{x}$, you can call it “the square root of $x$.” But there are other kinds of roots of $x$ too (which are indicated by little numbers tucked into the left side of the radical sign):

“The cube root of $x$” means the number that’s multiplied by itself three times in order to get $x$; “the fourth root of $x$” means the number that’s multiplied by itself four times in order to get $x$, and so on.

Since roots are the opposite operation of exponents, you can convert between roots and exponents. For example, taking the square root of $x$ is the same as raising $x$ to the $1/2$ power. To see this, apply the power rule for exponents:

$\left(x^\frac12\right)^2=x^{\left(\frac12\cdot2\right)}=x^1=x$

Here’s how to convert between roots and exponents.

Understanding the relationship between radicals and exponents

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How to simplify radicals

Example

Simplify the radical expression.

$\sqrt{9}$

We’re taking the square root of $9$, which means we need to figure out what number we have to multiply by itself in order to get $9$.

If we multiply $3$ by itself, we get $9$, which means that the square root of $9$ is $3$. So we can say

$\sqrt{9}=3$

If we’re given a number that $x$ stands for, we want $\sqrt{x}$ to represent only one number - and a number that everyone will agree on. Notice, however, that we can get $9$ not only by multiplying $3$ by itself, but also by multiplying $-3$ by itself:

$(+3)(+3)=9=(-3)(-3)$

Since roots are the opposite operation of exponents, you can convert between roots and exponents.

The way we get around this is that everyone agrees that both $\sqrt9$ and $9^\frac12$ mean the positive number that you can multiply by itself in order to get $9$:

$\sqrt9=3=9^\frac12$

Also

$\sqrt0=0=0^\frac12$

because $0$ is the only number that when multiplied by itself gives $0$. Now notice that there is no negative number that when multiplied by itself gives a negative number. (A positive number multiplied by itself is positive, $0$ multiplied by itself is $0$, and a negative number multiplied by itself is positive.) So $\sqrt{x}$ and $x^\frac12$ are undefined if $x$ is negative.

If $x$ stands for any positive number, there’s one and only one positive number that when multiplied by itself gives $x$. So $\sqrt{x}$ and $x^\frac12$ are defined, and they represent that “one and only one positive number.”

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In later sections, we’ll look at the specific rules we use to handle radical expressions.

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