Introduction to Radicals

Learn what radicals (square roots) are and how to simplify them.

Intermediate25 minLesson

Definition

A radical is a symbol that represents a root of a number. The most common radical is the square root, written with the radical symbol .

The square root of a number is a value that, when multiplied by itself, gives :

For example:

because
because
because

Parts of a radical:

The symbol is called the radical sign
The number under the radical is called the radicand
In , the radicand is 16

Try it now

What is ?

Worked Examples

Evaluate

Ask: What number times itself equals 49?

→ Looking for the square root

Think of perfect squares

→ 7 is the answer

Verify the answer

and → Confirmed!

Answer:

Common Mistakes

Thinking

Why it's wrong: You cannot split a square root over addition! , but . These are different!

Correct: Always simplify inside the radical first:

Forgetting that , not just

Why it's wrong: The square root always gives a non-negative result. For example, , not .

Correct: Remember: (absolute value)

Confusing with

Why it's wrong: Square root is NOT division by 2! (what times itself equals 16), but .

Correct: Square root asks: 'What number squared gives me this?'

Interactive Visual

Base:

Calculate:2^3 = 8

See how powers of a number grow on the number line. Change the base and exponent.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is ?

Why It Matters

Radicals appear throughout mathematics and real life:

Geometry: Finding the length of a diagonal or hypotenuse using the Pythagorean theorem:
Physics: Calculating distance, speed, or pendulum periods
Construction: Determining diagonal measurements for precise cuts
Finance: Calculating compound interest rates

Understanding radicals unlocks more advanced math, including quadratic equations and trigonometry!

Real World Applications

TV Screen Size

TV screens are measured diagonally. To find the diagonal, you use the Pythagorean theorem with radicals.

Example:

A TV has width 48 inches and height 36 inches. The diagonal is inches.

1Try It Yourself

A computer monitor has width 16 inches and height 12 inches.

What is the diagonal screen size?

Step 1: Write the mathematical expression

Find

Walking Distance

When you walk diagonally across a rectangular park, you can use radicals to find the shorter path.

Example:

A park is 300 meters by 400 meters. Walking diagonally: meters.

2Try It Yourself

A rectangular field is 60 meters long and 80 meters wide.

How far is it to walk diagonally across the field?

Step 1: Write the mathematical expression

Calculate

Key Takeaways

1A radical symbol () represents a root of a number
2The square root of is a number that, when squared, gives : if , then
3Perfect squares (1, 4, 9, 16, 25, 36, ...) have whole number square roots
4The number under the radical sign is called the radicand
5Square root is the inverse operation of squaring

Frequently Asked Questions

In the real number system, negative numbers do not have real square roots because no real number times itself gives a negative. However, in advanced math, we use imaginary numbers (like ) to handle this.

Glossary

Radical: A symbol () indicating a root of a number
Square root: A number that, when multiplied by itself, gives the original number
Radicand: The number or expression under the radical sign
Perfect square: A number that is the square of an integer (1, 4, 9, 16, 25, ...)
Radical sign: The symbol used to indicate a root

Formula Card

Square Root Definition

means

The square root of $n$ is the number that, when squared, equals $n$

Square and Square Root are Inverses

and

Square root undoes squaring (for non-negative numbers)

Perfect Squares Pattern

First ten perfect squares: $1^2, 2^2, 3^2, ..., 10^2$