Square Root Property Formula
Last Updated : 25 Dec, 2023
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