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Square Roots

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Square Roots

Study Guide

Key Definition

A square root of a number $b$ is a solution of the equation $x^2 = b$. In the real numbers, only positive numbers have two real square roots (a positive and a negative); zero has exactly one real square root; and negative numbers have no real square roots. The principal (non-negative) square root is written $\sqrt{b}$.

Important Notes

The principal square root is always non-negative.
Only positive real numbers have two real square roots; zero has exactly one; negative numbers have no real square roots.
The notation $±\sqrt{b}$ indicates both positive and negative roots.
Not all numbers have a whole number square root.
$\sqrt{2}$ is an example of an irrational number.
Estimation can be used to approximate square roots of non-perfect squares.

Mathematical Notation

$\sqrt{b}$ means the principal square root of $b$

$±\sqrt{b}$ represents both the positive and negative square roots

$x^2$ represents $x$ squared

$\frac{a}{b}$ represents a fraction

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

The principal square root undoes the operation of squaring a number. If $x^2 = b$ and $x ≥ 0$, then $\sqrt{b} = x$.

Remember

To find a square root, look for a number that, when squared, equals the original number.

Quick Reference

Perfect Squares:Examples: $4 = 2^2$, $9 = 3^2$

Estimation:If $64 < 70 < 81$, then $8 < \sqrt{70} < 9$

Understanding Square Roots

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A square root is a number that, when multiplied by itself, gives the original number. For example, $\sqrt{25} = 5$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the square root of $16$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are building a square garden with an area of $64$ square meters. What is the length of each side of the garden?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Estimate the square root of $50$. Hint: Find the nearest perfect squares.

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following is closest to $\sqrt{85}$?

Recap

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Review key concepts and takeaways