Square Root Property Formula

When one integer is multiplied by another integer, the resulting number is referred to as a square number. A number’s square root is that factor of a number that, when multiplied by itself, yields the original number. The square of a number a is denoted by a² and its square root is represented by the symbol √a. For example, the square of the number 4 is 4 × 4 = 16. But the square root of 4 is √4 = 2. 

Square Root Property Formula

There are certain properties or characteristics that need to be followed while solving square root expressions. 

Property 1: If two square root values have to be multiplied individually, they can be multiplied inside a common square root for the purpose of simplification, i.e., √p ⋅ √q = √(pq).

Example:

• √2 ⋅ √3 = √6
• √5 ⋅ √3 = √15

Property 2: If the square root of a fraction has to be evaluated, then the square roots of its numerator and denominator may be evaluated separately and then divided by each other, i.e., √(p/q) = √p/√q.

Example:

• √(13/5) = √13/√5
• √(5/12) = √5/√12

Property 3: Any two values cannot be added/subtracted together in a single square root if they are in separate roots, i.e., √a ± √b ≠ √(a ± b).

Example:

• √3 + √5 ≠ √8
• √4 + √7 ≠ √11

Property 4: If a value present in the square root is a perfect square, then it must be taken out of the root to simplify the calculations in the expression, i.e., √c²p = c√p.

Example:

• √2² ⋅ 5 = 2√5
• √6² ⋅ 7 = 6√7

Sample Problems

Problem 1. Simplify the expression √12 ⋅ √7 + √15/√147.

Solution:

We have the expression, √12 ⋅ √7 + √15/√147

Using the square root properties we have,

√12 ⋅ √7 + √15/√98 = √(2 ⋅ 2 ⋅ 3) ⋅ √7 + √15/√(3 ⋅ 7 ⋅ 7)

= 2 ⋅ √21 + (1/7) √15/√3

= 2√21 + (1/7) √(15/3)

= 2√21 + √5/7

Problem 2. Simplify the expression √219/√3 + √153 ⋅ √49.

Solution:

We have the expression, √219/√3 + √153 ⋅ √49

Using the square root properties we have,

√219/√3 + √153 ⋅ √49 = √(219/3) + √(3 ⋅ 3 ⋅ 17) ⋅ √(7 ⋅ 7)

= √73 + 3√17 ⋅ 7

= √73 + 21√17

Problem 3. Simplify the expression √415/√10 + √36/√9.

Solution:

We have the expression, √415/√10 + √36/√9

Using the square root properties we have,

√415/√10 + √36/√9 = √415/√10 + √(36/9)

= √(5 ⋅  83)/√10 + √(36/9)

= √(5 ⋅  83)/√10 + √4

= √83/√2 + 2

Problem 4. Simplify the expression √432 + √125/√5.

Solution:

We have the expression, √432 + √125/√5

Using the square root properties we have,

√432 + √125/√5 = √432 + √(125/5)

= √(3 ⋅ 144) + √(125/5)

= 3√12 + √25

= 3√(2 ⋅ 2 ⋅ 3) + 5 

= 6√3 + 5

Problem 5. Simplify the expression (√81/√9) (√196/√14).

Solution:

We have the expression, (√81/√9) (√196/√14)

Using the square root properties we have,

(√81/√9) (√196/√14) = √(81/9) √(196/14)

= √9 (√14)

= 3 √14

Problem 6. Simplify the expression (√225/√15)/(√9/√3).

Solution:

We have the expression, (√225/√15)/(√9/√3)

Using the square root properties we have,

(√225/√15)/(√9/√3) = (√(225/15)/√(9/3)

= √15/√3

= √(15/3)

= √5

Problem 7. Simplify the expression (√361/√19) – (√216/√36) + (√18/√3).

Solution:

We have the expression, (√361/√19) – (√216/√36) + (√18/√3)

Using the square root properties we have,

(√361/√19) – (√216/√36) + (√18/√3) = √(361/19) – √(216/36) – √(18/3)

= √19 – √6 – √6

= √19 – 2√6