Radicals and nth roots | Algebra 2

Undoing a power

Squaring $$9$$ gives $$81$$ . Going backwards, a square root of $$81$$ is a number whose square is $$81$$ . There are two: $$9$$ and $$-9$$ . The radical sign means the positive one, the principal square root:

\sqrt{81} = 9

The same idea works for any power. Since $$5^3 = 125$$ , the cube root of $$125$$ is $$5$$ : $\sqrt[3]{125} = 5$ .

Even and odd indices

Odd index. Every real number has exactly one real odd root, with the same sign: $\sqrt[3]{125} = 5$ and $\sqrt[3]{-125} = -5$ , since $$(-5)^3 = -125$$ .

Even index. An even power is never negative, so a negative number has no real even root: $\sqrt{-4}$ and $\sqrt[4]{-81}$ are not real numbers. A positive number has two even roots, and $\sqrt[n]{a}$ means the positive one: $\sqrt[4]{16} = 2$ , not $\pm 2$ .

The domain of a radical expression

When the radicand contains a variable, an even index forces a condition: radicand $\ge 0$ . An odd index needs no condition.

Simplifying the index

If the radicand is a power, you can divide the index and the exponent by a common factor (for $a \ge 0$ ):

\sqrt[n \cdot k]{a^{m \cdot k}} = \sqrt[n]{a^m}

The same rule works backwards, to give two radicals the same index.

Watch out: $\sqrt{x^2}$

It's tempting to say $\sqrt{x^2} = x$ . Try $$x = -3$$ :

\sqrt{(-3)^2} = \sqrt{9} = 3

The answer is $$3$$ , not $$-3$$ . A principal square root is never negative, so in general:

Comparing radicals

Which is bigger, $\sqrt[3]{3}$ or $\sqrt{2}$ ? Rewrite both with the same index (the LCM of the indices) and compare the radicands.

Practice

Quick check

What is $\sqrt[4]{16}$ ?