Lesson 1: Radicals

Property

If $b^n = a$, then $b$ is an $n^{\text{th}}$ root of $a$.
The principal $n^{\text{th}}$ root of $a$ is written $\sqrt[n]{a}$.
$n$ is called the index of the radical.
Properties of $\sqrt[n]{a}$
When $n$ is an even number and:

• $a \geq 0$, then $\sqrt[n]{a}$ is a real number.
• $a < 0$, then $\sqrt[n]{a}$ is not a real number.

When $n$ is an odd number, $\sqrt[n]{a}$ is a real number for all values of $a$.

Examples

• To simplify $\sqrt[3]{125}$, we look for a number that, when cubed, is 125. Since $5^3 = 125$, $\sqrt[3]{125} = 5$.

Property

Nested radicals can be simplified by converting to fractional exponents: $\sqrt[m]{\sqrt[n]{x^p}} = x^{\frac{p}{mn}}$

For the specific case: $\sqrt{\sqrt[3]{x^2}} = x^{\frac{2}{6}} = x^{\frac{1}{3}} = \sqrt[3]{x}$

Property

When the index of the radical is even, the radicand must be greater than or equal to zero.
When the index of the radical is odd, the radicand can be any real number.

Examples

• To find the domain of $f(x) = \sqrt{3x - 4}$, set the radicand $3x - 4 \geq 0$. Solving gives $x \geq \frac{4}{3}$. The domain is $[\frac{4}{3}, \infty)$.
• To find the domain of $f(x) = \sqrt[3]{2x^2 + 3}$, the index is odd, so the radicand can be any real number. The domain is $(-\infty, \infty)$.
• For $g(x) = \sqrt{\frac{6}{x-1}}$, the radicand must be positive. Since the numerator is positive, the denominator must be positive, so $x-1 > 0$. The domain is $(1, \infty)$.

Explanation

To find a radical function's domain, check the index. For an even index, the expression inside the radical must be non-negative. For an odd index, the domain includes all real numbers because odd roots can handle any value.