Learn on PengiBig Ideas Math, Algebra 1Chapter 6: Exponential Functions and Sequences

Lesson 2: Radicals and Rational Exponents

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.

Section 1

$n^{th}$ Root of a Number

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 $5^3 = 125$ , $\sqrt[3]{125} = 5$ .

Section 2

Roots of Negative Numbers

Property

Every positive number has two real-valued roots, one positive and one negative, if the index is even.
A negative number has no real-valued root if the index is even.
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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

$n^{th}$ Root of a Number

Property

$a \geq 0$ , 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 $5^3 = 125$ , $\sqrt[3]{125} = 5$ .

Section 2

Roots of Negative Numbers

Property

Every positive number has two real-valued roots, one positive and one negative, if the index is even.
A negative number has no real-valued root if the index is even.
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.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

$n^{th}$ Root of a Number

Property

$a \geq 0$ , 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 $5^3 = 125$ , $\sqrt[3]{125} = 5$ .

Section 2

Roots of Negative Numbers

Property

Every positive number has two real-valued roots, one positive and one negative, if the index is even.
A negative number has no real-valued root if the index is even.
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.