Study Guide
Introduction to Square Roots
A square root of a number x is a value that, when multiplied by itself, gives x. For example, 49 = 7
because 7 × 7 = 49. While some numbers are perfect squares, many others (like 50) need to be
simplified into a mix of a whole number and a radical.
The Perfect Square Method
To simplify n, find the largest perfect square factor of n.
1. Identify Factors: List the perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, …
2. Rewrite: Write the radical as a product: Perfect Square ⋅ Remainder.
3. Extract: Take the square root of the perfect square and place it outside the radical symbol.
Example: Simplify 75
The largest perfect square factor of 75 is 25.
Rewrite: 25 ⋅ 3
Extract: 5 3
The Factor Tree (Pair) Method
If you can’t find a perfect square factor, use prime factorization.
1. Factor Tree: Break the number down into its prime factors.
2. Find Pairs: Look for pairs of the same number.
3. Outside/Inside Rule: For every pair, one number goes outside the radical. Any numbers without a
partner stay inside and are multiplied together.
Example: Simplify 24
Prime factors: 2, 2, 2, 3
One pair of 2s, with a 2 and a 3 left over.
Result: 2 2 ⋅ 3 = 2 6
Multiplying Radicals (The FOIL Method)
When multiplying expressions like (a + b)(c +
d) , use the FOIL method:
First terms
Outer terms
Inner terms
Last terms
Introduction to Square Roots
A square root of a number x is a value that, when multiplied by itself, gives x. For example, 49 = 7
because 7 × 7 = 49. While some numbers are perfect squares, many others (like 50) need to be
simplified into a mix of a whole number and a radical.
The Perfect Square Method
To simplify n, find the largest perfect square factor of n.
1. Identify Factors: List the perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, …
2. Rewrite: Write the radical as a product: Perfect Square ⋅ Remainder.
3. Extract: Take the square root of the perfect square and place it outside the radical symbol.
Example: Simplify 75
The largest perfect square factor of 75 is 25.
Rewrite: 25 ⋅ 3
Extract: 5 3
The Factor Tree (Pair) Method
If you can’t find a perfect square factor, use prime factorization.
1. Factor Tree: Break the number down into its prime factors.
2. Find Pairs: Look for pairs of the same number.
3. Outside/Inside Rule: For every pair, one number goes outside the radical. Any numbers without a
partner stay inside and are multiplied together.
Example: Simplify 24
Prime factors: 2, 2, 2, 3
One pair of 2s, with a 2 and a 3 left over.
Result: 2 2 ⋅ 3 = 2 6
Multiplying Radicals (The FOIL Method)
When multiplying expressions like (a + b)(c +
d) , use the FOIL method:
First terms
Outer terms
Inner terms
Last terms
