Introduction to Radicals

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

Intermediate25 minLesson
Printable worksheet

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 x\sqrt{\phantom{x}}.
The square root of a number nn is a value that, when multiplied by itself, gives nn:
n=x means x2=n\sqrt{n} = x \text{ means } x^2 = n
For example:
  • 9=3\sqrt{9} = 3 because 32=93^2 = 9
  • 25=5\sqrt{25} = 5 because 52=255^2 = 25
  • 100=10\sqrt{100} = 10 because 102=10010^2 = 100
Parts of a radical:
  • The symbol x\sqrt{\phantom{x}} is called the radical sign
  • The number under the radical is called the radicand
  • In 16\sqrt{16}, the radicand is 16

Try it now

What is 16\sqrt{16}?

Worked Examples

Evaluate 49\sqrt{49}
1

Ask: What number times itself equals 49?

?×?=49? \times ? = 49 → Looking for the square root
2

Think of perfect squares

7×7=497 \times 7 = 49 → 7 is the answer
3

Verify the answer

72=497^2 = 49 and 49=7\sqrt{49} = 7 → Confirmed!

Common Mistakes

Thinking 9+16=9+16\sqrt{9 + 16} = \sqrt{9} + \sqrt{16}

Why it's wrong: You cannot split a square root over addition! 9+16=25=5\sqrt{9 + 16} = \sqrt{25} = 5, but 9+16=3+4=7\sqrt{9} + \sqrt{16} = 3 + 4 = 7. These are different!

Correct: Always simplify inside the radical first: 9+16=25=5\sqrt{9 + 16} = \sqrt{25} = 5

Forgetting that x2=∣x∣\sqrt{x^2} = |x|, not just xx

Why it's wrong: The square root always gives a non-negative result. For example, (−3)2=9=3\sqrt{(-3)^2} = \sqrt{9} = 3, not −3-3.

Correct: Remember: x2=∣x∣\sqrt{x^2} = |x| (absolute value)

Confusing 16\sqrt{16} with 16÷216 \div 2

Why it's wrong: Square root is NOT division by 2! 16=4\sqrt{16} = 4 (what times itself equals 16), but 16÷2=816 \div 2 = 8.

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

Interactive Visual

3² = 9
3

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

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

16 problems
Problem 1 of 16
Easy
What is 16\sqrt{16}?

Why It Matters

Radicals appear throughout mathematics and real life:
  • Geometry: Finding the length of a diagonal or hypotenuse using the Pythagorean theorem: c=a2+b2c = \sqrt{a^2 + b^2}
  • 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 482+362=2304+1296=3600=60\sqrt{48^2 + 36^2} = \sqrt{2304 + 1296} = \sqrt{3600} = 60 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 162+122\sqrt{16^2 + 12^2}

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: 3002+4002=90000+160000=250000=500\sqrt{300^2 + 400^2} = \sqrt{90000 + 160000} = \sqrt{250000} = 500 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 602+802\sqrt{60^2 + 80^2}

Key Takeaways

  • 1A radical symbol (x\sqrt{\phantom{x}}) represents a root of a number
  • 2The square root of nn is a number that, when squared, gives nn: if n=x\sqrt{n} = x, then x2=nx^2 = n
  • 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 i=−1i = \sqrt{-1}) to handle this.
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 i=−1i = \sqrt{-1}) to handle this.
16=4\sqrt{16} = 4 (just the positive root). But when solving x2=16x^2 = 16, the answer is x=±4x = \pm 4 because both 42=164^2 = 16 and (−4)2=16(-4)^2 = 16.
Numbers like 1, 4, 9, 16, 25 are called perfect squares because they can form a perfect square shape. For example, 9 = 3 rows of 3 dots, forming a square.

Glossary

Radical
A symbol (x\sqrt{\phantom{x}}) 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 x\sqrt{\phantom{x}} used to indicate a root

Formula Card

Square Root Definition

n=x\sqrt{n} = x means x2=nx^2 = n

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

Square and Square Root are Inverses

x2=∣x∣\sqrt{x^2} = |x| and (x)2=x(\sqrt{x})^2 = x

Square root undoes squaring (for non-negative numbers)

Perfect Squares Pattern

1,4,9,16,25,36,49,64,81,1001, 4, 9, 16, 25, 36, 49, 64, 81, 100

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

Today's challengeOne problem, three tries

More in This Topic

Simplifying Square Roots

Intermediate25 min

Multiplying Radicals

Intermediate25 min

Dividing Radicals

Intermediate25 min

Adding and Subtracting Radicals

Intermediate25 min