Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 1: Follow the Rules

Lesson 8: Radicals

In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn how to work with radicals, including evaluating square roots, cube roots, and higher-order roots using radical notation and fractional exponents. Students practice simplifying expressions like the square root of 12 into simplified form using prime factorization and the laws of exponents. The lesson also covers the general rule that the square root of a nonnegative number is always nonnegative, and introduces the identity x to the m/n power equals the nth root of x to the m.

Section 1

Square Root of a Number

Property

Square of a Number
If $n^2 = m$ , then $m$ is the square of $n$ .

Square Root of a Number
If $n^2 = m$ , then $n$ is a square root of $m$ .

Square Root Notation
$\sqrt{m}$ is read as “the square root of $m$ .”
If $m = n^2$ , then $\sqrt{m} = n$ , for $n \ge 0$ .
The square root of $m$ , $\sqrt{m}$ , is the positive number whose square is $m$ . This is also called the principal square root.
To find the negative square root of a number, we place a negative in front of the radical sign.
When using the order of operations, we treat the radical as a grouping symbol.

Section 2

Square Root of Squared Variables

Property

For any real number $a$ : $\sqrt{a^2} = |a|$

This means $\sqrt{a^2} = a$ only when $a \geq 0$ , and $\sqrt{a^2} = -a$ when $a < 0$ .

Section 3

Rational Exponents

Property

Exponential Notation for Radicals.
For any integer $n \geq 2$ and for $a \geq 0$ ,
$a^{1/n} = \sqrt[n]{a}$

Rational Exponents.
$a^{m/n} = (a^{1/n})^m = (a^m)^{1/n}$ , $a > 0$ , $n \neq 0$

Rational Exponents and Radicals.
$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Square Root of a Number

Property

Square of a Number
If $n^2 = m$ , then $m$ is the square of $n$ .

Square Root of a Number
If $n^2 = m$ , then $n$ is a square root of $m$ .

Section 2

Square Root of Squared Variables

Property

For any real number $a$ : $\sqrt{a^2} = |a|$

This means $\sqrt{a^2} = a$ only when $a \geq 0$ , and $\sqrt{a^2} = -a$ when $a < 0$ .

Section 3

Rational Exponents

Property

Exponential Notation for Radicals.
For any integer $n \geq 2$ and for $a \geq 0$ ,
$a^{1/n} = \sqrt[n]{a}$

Rational Exponents.
$a^{m/n} = (a^{1/n})^m = (a^m)^{1/n}$ , $a > 0$ , $n \neq 0$

Rational Exponents and Radicals.
$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Overview

Lesson overview

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

Overview

Section 1

Square Root of a Number

Property

Square of a Number
If $n^2 = m$ , then $m$ is the square of $n$ .

Square Root of a Number
If $n^2 = m$ , then $n$ is a square root of $m$ .

Section 2

Square Root of Squared Variables

Property

For any real number $a$ : $\sqrt{a^2} = |a|$

This means $\sqrt{a^2} = a$ only when $a \geq 0$ , and $\sqrt{a^2} = -a$ when $a < 0$ .

Section 3

Rational Exponents

Property

Exponential Notation for Radicals.
For any integer $n \geq 2$ and for $a \geq 0$ ,
$a^{1/n} = \sqrt[n]{a}$

Rational Exponents.
$a^{m/n} = (a^{1/n})^m = (a^m)^{1/n}$ , $a > 0$ , $n \neq 0$

Rational Exponents and Radicals.
$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$