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Simplified Radical Form - Resource Sheet, Exercises of Mathematics

Mathematics

Resource sheet for understanding how to write numbers in simplified radical form.

Typology: Exercises

2025/2026

Uploaded on 04/18/2026

thien-chu
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Topic Resource โ€“ Simplified Radical Form
VOCABULARY/TERMINOLOGY
Term Definition Example
Factor
A positive integer that divides a
number exactly, leaving no
remainder
Factors of 36 = 1, 2, 3, 6, 12,
18, 36
Radical
A symbol that represents the
square root or nth root of a
number; โˆš
โˆš
36
= 6;
โˆš
49
= 7
Square Root
The square root of a number is a
number that, when multiplied by
itself, gives the original term
โˆš
36
= 6; 6 ร— 6 = 36
Perfect
Square
A number whose square root is a
whole number
โˆš
25
= 5 โ€“ perfect square
โˆš
40
โ‰ˆ 6.32 โ€“ non-perfect square
Rather than write non-perfect square roots as decimals, we can express them in a more
simplified radical form. The square root symbol (โˆš), also known as a radical, represents
a grouping of terms. Numbers under a radical can be factored just like numbers in
parentheses.
Examples:
Notice that in some of the examples for radical factoring, we are left with the square
roots of perfect squares. Perfect square roots can be simplified and written outside of
the radical (like a coefficient) as its square root value. This is often referred to as
simplified radical form.
HOW TO WRITE RADICALS IN SIMPLIFIED RADICAL FORM โ€“
Example 1:
Write the number
โˆš
48
in simplified radical form.
Steps to solve
1. Write out all factors of the number
under the radical symbol
Factors of 48 = 1, 2, 4, 6, 8, 16,
24, 48
2. Determine the factor of the number
that is the greatest perfect square
16 is the greatest perfect square
factor of 48
3. Write the square root as a factor
pair using the greatest perfect
square
โˆš
48
=
โˆš
16
ร—
โˆš
3
Parentheses Factoring Radical Factoring
(80) = (8) ร— (10)
โˆš
80
=
โˆš
8
ร—
โˆš
10
(24) = (4) ร— (6) = (8)
ร— (3)
โˆš
24
=
ร—
โˆš
6
=
โˆš
8
ร—
โˆš
3
(75) = (25) ร— (3)
โˆš
75
=
โˆš
25
ร—
โˆš
3
pf3

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Topic Resource โ€“ Simplified Radical Form VOCABULARY/TERMINOLOGY Term Definition Example Factor A positive integer that divides a number exactly, leaving no remainder

Factors of 36 = 1, 2, 3, 6, 12,

Radical A symbol that represents the square root or nth^ root of a number; โˆš

โˆš^36 = 6;^ โˆš^49 = 7

Square Root The square root of a number is a number that, when multiplied by itself, gives the original term

โˆš^36 = 6; 6 ร— 6 = 36

Perfect Square A number whose square root is a whole number

โˆš 25 = 5 โ€“ perfect square

โˆš 40 โ‰ˆ 6.32 โ€“ non-perfect square

Rather than write non-perfect square roots as decimals, we can express them in a more simplified radical form. The square root symbol (โˆš), also known as a radical, represents a grouping of terms. Numbers under a radical can be factored just like numbers in parentheses. Examples: Notice that in some of the examples for radical factoring, we are left with the square roots of perfect squares. Perfect square roots can be simplified and written outside of the radical (like a coefficient) as its square root value. This is often referred to as simplified radical form. HOW TO WRITE RADICALS IN SIMPLIFIED RADICAL FORM โ€“

Example 1: Write the number โˆš 48 in simplified radical form.

Steps to solve

  1. Write out all factors of the number under the radical symbol Factors of 48 = 1, 2, 4, 6, 8, 16, 24, 48
  2. Determine the factor of the number that is the greatest perfect square 16 is the greatest perfect square factor of 48
  3. Write the square root as a factor pair using the greatest perfect square

โˆš^48 =^ โˆš^16 ร—^ โˆš^3

Parentheses Factoring Radical Factoring

(80) = (8) ร— (10) โˆš 80 = โˆš 8 ร— โˆš 10

(24) = (4) ร— (6) = (8)

ร— (3)

โˆš 24 = โˆš 4 ร— โˆš 6 = โˆš 8 ร— โˆš 3

(75) = (25) ร— (3) โˆš 75 = โˆš 25 ร— โˆš 3

  1. Simplify the greatest perfect square factor, writing it as a coefficient

โˆš^48 =^^4 โˆš^3

  1. Double check to see if the number under the radical has any perfect square factors; if it does, repeat steps 1- 3 has no perfect square factors

Example 2a: Write the number โˆš 200 in simplified radical form.

Steps to solve

  1. Write out all factors of the number under the radical symbol Factors of 200 = 1, 2, 4, 5, 8, 10, 20, 25 , 40, 50, 100, 200
  2. Determine the factor of the number that is the greatest perfect square 25 is the greatest perfect square factor of 200*
  3. Write the square root as a factor pair using the greatest perfect square

โˆš^200 =^ โˆš^25 ร—^ โˆš^8

  1. Simplify the greatest perfect square factor, writing it as a coefficient

โˆš^200 =^^5 โˆš^8

  1. Double check to see if the number under the radical has any perfect square factors; if it does, repeat steps 1-4 with that number 8 has 4 as a perfect square factor We will remember 5 as a coefficient that has already been simplified

Repeating step 1 Factors of 8 = 1, 2, 4, 8

Repeating step 2

4 is the greatest perfect square factor

Repeating step 3 โˆš 8 = โˆš 4 ร— โˆš 2

Repeating step 4 โˆš 8 = 2โˆš 2

  1. If repeating steps 1-4, once you have simplified the radical, multiply any perfect square coefficients that were simplified from the original problem

5 ร— 2โˆš 2 = 10 โˆš 2

*In this example, we mistakenly pick 25 as the largest perfect square when 100 is also a perfect square of 200 and is greater; this is corrected in step 5 and addressed in step 6.

Example 2b: Write the number โˆš 200 in simplified radical form (re-visiting

for correctness)

Steps to solve

  1. Write out all factors of the number under the radical symbol Factors of 200 = 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100 , 200
  2. Determine the factor of the number 100 is the greatest perfect