The square root of 252 is irrational. The approximate value of square root of 252 is 15.87. A number “s” becomes a square root, when s × s gives a product p. “p” will be called the perfect square number and “s” its square root. In simple terms, when a number “s” is multiplied with itself, the product ”p” is called the perfect square and “s” its square root. “s” will be an integer as the product of -s × -s = s × s.
Consider the number 15 × 15 = 225 and 16 × 16 = 256. 252 lies between 225 and 256. So the square root of 252 lies between 15 and 16 as 252 is not a perfect square number. Hence, the square root of 252 is also not a perfect integer. Therefore, √252 is an irrational number. You can refer to Squares and Square roots.
Note the Following:
- The Square root of 252 = √252 where ‘√’ is the radical, and 252 is the radicand.
- Exponential Form of Square root of 252 = 2521/2
- Solution for √252 ≈ 15.87
- The square root of 252 is Irrational = True
What is the Square root of 252?
The square root of 252 is 15.87, as the square root of 252 is irrational.
|
√252 = 6√7 =15.87 |
How to Find the Square root of 252?
There are three methods to find the Square root of 252
- Prime Factorisation method
- Long Division method
- Repeated Subtraction method
Square root of 252 by Prime Factorisation Method
In the prime factorization method, the following steps are followed.
- Divide the given number with prime factors
- The 2 same prime factors are grouped.
- The product of the groups (considering only 1 from the group for the product) forms the square root for the required number.
Let us divide the number 252 by the prime factors, starting with the smallest, i.e. 2.
The given number, 252 will be expressed as;
|
2 |
252 |
|
2 |
126 |
|
3 |
63 |
|
3 |
21 |
|
7 |
7 |
|
× |
1 |
252 = 2 × 2 × 3 × 3 × 7
Grouping into 2 with same divisors:
Group 1 = 2 × 2, considering only 2
Group 2 = 3 × 3, considering only 3
Another divisor 7 is left out as it does not have a pair.
Therefore the square root of 252 = 2 × 3 × √7 = 6√7.
Square root of 252 by Long Division Method
To understand the long division method, follow the below mentioned detailed steps;
Step 1: Grouping the given number into pairs.
Step 2: Consider the first number, use the divisor such that the product is less than or equal to the first number. The divisors should be:
1 × 1 = 1
2 × 2 = 4
3 × 3 = 9
Step 3: Continue the division, by bringing the next pair
Step 4: The first divisor in the next division to be (Divisor of first division + quotient of first division) such that:
d1 × 1 =
d2 × 2 =
d3 x 3 =
And so on.
Step 5: If the division is complete with the remainder as zero, then the quotient will be the square root. Else the number is not a perfect square number.
Applying the above steps to 252
Groups are 2 and 52
The squares are:
1 × 1 = 1
2 × 2 = 4
No further division is possible and the remainder is non-zero. Hence 252 does not have a perfect square. Therefore, 252 is an irrational number which is 6√7.
Square root of 252 by Repeated Subtraction Method.
In the repeated Subtraction method, 252 is subtracted by 1 and the resultant by 3. This process of repeatedly subtracting by odd numbers until zero is obtained. If zero is obtained, then the number is a perfect square and the step at which it becomes zero is the square root. If the result is non-zero, then the number is not perfect and does not have a perfect square root.
The table below shows repeated subtraction for the given number 252.
|
Step 1 |
252 |
– |
1 |
= |
251 |
|
Step 2 |
251 |
– |
3 |
= |
248 |
|
Step 3 |
248 |
– |
5 |
= |
243 |
|
Step 4 |
243 |
– |
7 |
= |
236 |
|
Step 5 |
236 |
– |
9 |
= |
227 |
|
Step 6 |
227 |
– |
11 |
= |
216 |
|
Step 7 |
216 |
– |
13 |
= |
203 |
|
Step 8 |
203 |
– |
15 |
= |
188 |
|
Step 9 |
188 |
– |
17 |
= |
171 |
|
Step 10 |
171 |
– |
19 |
= |
152 |
|
Step 11 |
152 |
– |
21 |
= |
131 |
|
Step 12 |
131 |
– |
23 |
= |
108 |
|
Step 13 |
108 |
– |
25 |
= |
83 |
|
Step 14 |
83 |
– |
27 |
= |
56 |
|
Step 15 |
56 |
– |
29 |
= |
27 |
|
Step 16 |
27 |
– |
31 |
= |
-4 |
Since the difference does not become zero, 252 is not a perfect square number and hence it does not have a perfect square root. The root lies between the numbers 15 and 16, resulting in a decimal root.
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Solved Examples
1. Which number that is lesser than or greater than 252, that has a perfect square root?
225, which is less than 252 and 256 which is greater than 252, are the perfect square numbers. 15 and 16 are their perfect square roots, respectively.
2. What is the square root of 256?
16 is the square root of 256.
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