Lesson 2: Roots and Radicals

New Concept

This lesson connects roots to fractional exponents, showing that a radical like $\sqrt[n]{a}$ is the same as the power $a^{1/n}$. This notation simplifies solving radical equations and working with power functions involving roots.

What’s next

Next, you'll practice converting between radical and exponential notations. Soon, you'll apply these skills to solve equations in a series of challenge problems.

Property

$s$ is called an $n$th root of $b$ if $s^n = b$. We use the symbol $\sqrt[n]{b}$ to denote the $n$th root of $b$. An expression of the form $\sqrt[n]{b}$ is called a radical, $b$ is called the radicand, and $n$ is called the index of the radical.

Examples

• $\sqrt[3]{27} = 3$ because $3^3 = 27$.

• $\sqrt[4]{256} = 4$ because $4^4 = 256$.

Property

For any integer $n \geq 2$ and for $a \geq 0$,

Examples

• Write $64^{1/3}$ using radical notation and evaluate: $64^{1/3} = \sqrt[3]{64} = 4$.

• Write $\sqrt[4]{625}$ using exponential notation and evaluate: $\sqrt[4]{625} = 625^{1/4} = 5$.

Property

To solve an equation involving a power function $x^n$, first isolate the power, then raise both sides to the exponent $\frac{1}{n}$. This uses the third law of exponents, $(x^a)^b = x^{ab}$, to cancel the original exponent.
If $x^n = c$, then $(x^n)^{1/n} = c^{1/n}$, which simplifies to $x = c^{1/n}$.

Examples

• To solve $x^3 = 125$, raise both sides to the $1/3$ power: $(x^3)^{1/3} = 125^{1/3}$, so $x = 5$.

• To solve $2m^4 = 32$, first isolate the power: $m^4 = 16$. Then raise both sides to the $1/4$ power: $(m^4)^{1/4} = 16^{1/4}$, so $m = 2$.

Property

A radical equation is one in which the variable appears under a radical, which can be denoted by a fractional exponent. To solve, first isolate the radical expression. Then, raise both sides of the equation to the power equal to the index of the radical.
If $\sqrt[n]{x} = c$, then $(\sqrt[n]{x})^n = c^n$, which simplifies to $x=c^n$.

Examples

• To solve $\sqrt[3]{x} = 4$, cube both sides: $(\sqrt[3]{x})^3 = 4^3$, which gives $x = 64$.

• To solve $5x^{1/2} = 20$, first isolate the power: $x^{1/2} = 4$. Then square both sides: $(x^{1/2})^2 = 4^2$, so $x = 16$.

Property

1. Every positive number has two real-valued roots, one positive and one negative, if the index is even.
2. A negative number has no real-valued root if the index is even.
3. Every real number, positive, negative, or zero, has exactly one real-valued root if the index is odd.

The symbol $\sqrt[n]{b}$ refers to the principal (positive) root when $n$ is even.

Examples

• $\sqrt[3]{-64} = -4$ because $(-4)^3 = -64$. This is an odd root of a negative number.

• $\sqrt[4]{-81}$ is not a real number because the index (4) is even and the radicand (-81) is negative.