Square roots
Study the following situations.
(a) Area of a square is 144
We know that the area of a square =
If we assume the length of the side to be ‘a’, then
To find the length of side it is necessary to find a number whose square is 144.
(b) What is the length of a diagonal of a square of side 8 cm ?
Can we use Pythagoras theorem to solve this ? We have:
Again to get AC, we need to think of a number whose square is 128.
(c) In a right triangle, the length of the hypotenuse and a side are respectively, 5 cm and 3 cm. Can you find the third side ?
Using Pythagoras theorem
Again, to find 'x' we need a number whose square is 16.
In all the above cases, we need to find a number whose square is known. Finding the number with the known square is known as finding the square root.
Finding square roots
The inverse (opposite) operation of addition is
Similarly, finding the square root is the inverse operation of squaring a number.
We have:
Since
We say that square roots of 81 are
THINK, DISCUSS AND WRITE
From the above, you may say that there are two integral square roots of a perfect square number. In this chapter, we shall take up only positive square root of a natural number.
Positive square root of a number is denoted by the symbol
For example:
| Statement | Inference |
|---|---|
| Statement | Inference |
|---|---|
Finding square root through repeated subtraction
Do you remember that the sum of the first n odd natural numbers is
Consider
81 – 1 =
80 – 3 =
77 – 5 =
Continue this pattern
81 -1, 80 -3,
Pattern: “Successively subtract 1, 3, 5, 7, 9, ... from it, to get the next one.”
From 81 we have subtracted successive odd numbers starting from 1 and obtained 0 at 9th step. Therefore
Can you find the square root of 729 using this method? It is possible but it will be time consuming. Thus, we need a simpler way to find the square roots.
TRY THESE
By repeated subtraction of odd numbers starting from 1, find whether the following numbers are perfect squares or not? If the number is a perfect square then find its square root.
(i)
(i) 121
81 – 1 =
80 – 3 =
77 – 5 =
72 – 7 =
65 – 9 =
56 – 11 =
45 – 13 =
32 – 15 =
17 – 17 =
Yes, it is a perfect square.
(ii)
(ii) 55
55 – 1 =
54 – 3 =
51 – 5 =
46 – 7 =
39 – 9 =
30 – 11 =
19 – 13 =
6 – 15 =
No, it isn't a perfect square.
(iii)
(iii) 36
36 – 1 =
35 – 3 =
32 – 5 =
27 – 7 =
20 – 9 =
11 – 11 =
Yes, it is a perfect square.
(iv)
(iv) 49
49 – 1 =
48 – 3 =
45 – 5 =
40 – 7 =
33 – 9 =
24 – 11 =
13 – 13 =
Yes, it is a perfect square.
(v)
(v) 90
90 – 1 =
89 – 3 =
86 – 5 =
81 – 7 =
74 – 9 =
65 – 11 =
54 – 13 =
41 – 15 =
26 – 17 =
9 – 15 =
No, it isn't a perfect square.
Finding square root through prime factorisation
Consider the prime factorisation of the following numbers and their squares:
| Prime factorisation of a Number | Prime factorisation of its Square |
|---|---|
| 6 = | 36 = 2 × 2 × 3 × 3 |
| 8 = | 64 = 2 × 2 × 2 × 2 × 2 × 2 |
| 12 = 2 × 2 × 3 | |
| 15 = 3 × 5 | 225 = |
How many times does 2 occur in the prime factorisation of 6?
How many times does 2 occur in the prime factorisation of 36?
Similarly, observe the occurrence of 3 in 6 and 36, of 2 in 8 and 64 etc.
You will find that each prime factor in the prime factorisation of the square of a number, occurs twice the number of times it occurs in the prime factorisation of the number itself.
Let us use this to find the square root of a given square number, say 324.
324 | ||||||||||||
2 | × | 162 | ||||||||||
2 | × | 81 | ||||||||||
3 | × | 27 | ||||||||||
3 | × | 9 | ||||||||||
3 | × | 3 | ||||||||||
324 | = | 2 | × | 2 | × | 3 | × | 3 | × | 3 | × | 3 |
We know that the prime factorisation of 324 is
324 =
By pairing the prime factors, we get
324 = 2 × 2 × 3 × 3 × 3 × 3 =
So,
Similarly can you find the square root of 256? Prime factorisation of 256 is:
256 | ||||||||||||||
2 | × | 128 | ||||||||||||
2 | × | 64 | ||||||||||||
2 | × | 32 | ||||||||||||
2 | × | 16 | ||||||||||||
2 | × | 8 | ||||||||||||
2 | × | 4 | ||||||||||||
2 | × | 2 | ||||||||||||
256 | = | 2 | × | 2 | × | 2 | × | 2 | × | 2 | × | 2 | × | 2 |
256 =
By pairing the prime factors we get,
256 = (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) =
Therefore,
Is 48 a perfect square?
We know,
48 = (2 × 2) × (2 × 2) × 3
Since all the factors are not in pairs so 48 is not a perfect square.
Suppose we want to find the smallest multiple of 48 that is a perfect square, how should we proceed?
Making pairs of the prime factors of 48 we see that
So we need to multiply by 3 to complete the pair.
Hence:
48 × 3 =
Can you tell by which number should we divide 48 to get a perfect square?
The factor
So, if we divide 48 by 3 we get 48 ÷ 3 =
and this number 16 is a perfect square too.
Example 4: Find the square root of 6400.
Writing prime factorisation for 6400: 6400 =
Therefore,
Example 5: Is 90 a perfect square?
90 | ||||||||
2 | × | 45 | ||||||
3 | × | 15 | ||||||
3 | × | 5 | ||||||
90 | = | 2 | × | 3 | × | 3 | × | 5 |
We have: 90 =
The prime factors 2 and 5 do not occur in pairs.
Therefore, 90 is not a
That 90 is not a perfect square can also be seen from the fact that it has only one zero.
Example 6: Is 2352 a perfect square? If not, find the smallest multiple of 2352 which is a perfect square. Find the square root of the new number.
We have: 2352 =
As the prime factor 3 has no pair, 2352 is not a perfect square.
If 3 gets a pair then the number will become perfect square. So, we multiply 2352 by 3 to get,
2352 × 3 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
Now each prime factor is in a pair.
Therefore, 2352 × 3 =
Thus, the required smallest multiple of 2352 is 7056 which is a perfect square.
Example 7: Find the smallest number by which 9408 must be divided so that the quotient is a perfect square. Find the square root of the quotient.
We have, 9408 =
If we divide 9408 by the factor 3, then
9408 ÷ 3 =
which is a perfect square.
Therefore, the required smallest number by which 9408 needs to be divided, to become a perfect square, is
And,
3136 =
Example 8: Find the smallest square number which is divisible by each of the numbers 6, 9 and 15.
This has to be done in two steps. First, find the smallest common multiple and then find the square number needed. The least number divisible by each one of 6, 9 and 15 will be their
The LCM of 6, 9 and 15 is
We see that prime factors 2 and 5 are not in pairs.
Therefore, 90 is not a perfect square.
In order to get a perfect square, each factor of 90 must be paired. So we need to make pairs of
Therefore, 90 should be multiplied by
Hence, the required square number is 90 × 10 =
Prime Factorization Finder
Finding square root by division method
When the numbers are large, even the method of finding square root by prime factorisation becomes lengthy and difficult. To overcome this problem we use Long Division Method.
For this we need to determine the number of digits in the square root. See the following table:
| Number | Square | Property |
|---|---|---|
| 10 | 100 | which is the smallest |
| 31 | 961 | which is the greatest |
| 32 | 1024 | which is the smallest |
| 99 | 9801 | which is the greatest |
So, what can we say about the number of digits in the square root if a perfect square is a 3-digit or a 4-digit number? We can say that, if a perfect square is a 3-digit or a 4-digit number, then its square root will have
Can you tell the number of digits in the square root of a 5-digit or a 6-digit perfect square?
- The smallest 5-digit perfect square is
10000 which gives10000 while the largest 5-digit perfect square is 99856 which gives99856 - This tells us that the square root has
digits. - Checking for 6-digit perfect square: we have
100489 which gives100489 which is the smallest perfect square while the largest 5-digit perfect square is 998001 which gives998001 - This tells us that in this case as well, the square root has
digits. - Thus, the square root of 5 (or) 6-digit perfect square has a total of 3 digits.
The smallest 3-digit perfect square number is 100 which is the square of 10 and the greatest 3-digit perfect square number is 961 which is the square of 31. The smallest 4-digit square number is 1024 which is the square of 32 and the greatest 4-digit number is 9801 which is the square of 99.
Think Discuss and Write
Can we say that if a perfect square is of n-digits, then its square root will have
The relationship between the number of digits in a perfect square and the number of digits in its square root follows a specific pattern, but it does not fit perfectly into the formulas
If n is even: A perfect square with
Example:
10000 (a 5-digit perfect square) has
1000000 (a 7-digit perfect square) has
If n is odd:
A perfect square with n digits typically has
The formula you suggested is generally correct but can be considered an approximation. The exact number of digits can vary slightly depending on the specific numbers involved.
The use of the number of digits in square root of a number is useful in the following method: Consider the following steps to find the square root of 529.
- Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 5 29
- From the left, find the square which is less than or equal to the digit. Here we have, (
2 2 3 2 when subtracting 2 2 - Bring down the number under the next bar (i.e. 29) to the right of the remainder. We now get the new dividend i.e.
- In the divisor put the first digit as double the quotient and place a blank its right. i.e. 4_
- Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
- As
× 3 = 129, we can choose the digit to be filled in the blank as . - Since, the remainder is
and there are no more digits left in the given number, we get: 529 - We have found the square root of 529.
Now consider
- Place a bar over every pair of digits starting from the one’s digit. (40 96)
- From the left, find the square which is less than or equal to the digit. Here we have, (
6 2 7 2 when subtracting 6 2 - Bring down the number under the next bar (i.e. 96) to the right of the remainder. We now get the new dividend i.e.
- In the divisor put the first digit as double the quotient and place a blank its right. i.e. 12_
- Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
- We get that: 124 ×
= 496 - Since, the remainder is
and there are no more digits left in the given number, we get: 4096 . - We have found the square root of 4096.
Estimating the number
We use bars to find the number of digits in the square root of a perfect square number.
= 23 and = 64
In both the numbers 529 and 4096 there are two bars and the number of digits in their square root is
Can you tell the number of digits in the square root of 14400 ? By placing bars on top of , we get
Thus, the square root will be having 3 digits.
Without calculating square roots, find the number of digits in the square root of the following numbers.
Example 9: Find the square root of:
(i) 729
- Place a bar over every pair of digits starting from the one’s digit. (7 29)
- From the left, find the square which is less than or equal to the digit. Here we have, (
2 2 3 2 when subtracting 2 2 - Bring down the number under the next bar (i.e. 29) to the right of the remainder. We now get the new dividend i.e.
- In the divisor put the first digit as double the quotient and place a blank its right. i.e. 4_
- Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
- We get that: 47 ×
= 329. - Since, the remainder is
and there are no more digits left in the given number, we get: 729 . - We have found the square root of 729.
(ii) 1296
- Place a bar over every pair of digits starting from the one’s digit. (12 96)
- From the left, find the square which is less than or equal to the digit. Here we have, (
3 2 4 2 when subtracting 3 2 - Bring down the number under the next bar (i.e. 96) to the right of the remainder. We now get the new dividend i.e.
- In the divisor put the first digit as double the quotient and place a blank its right. i.e. 6_
- Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
- We get that: 66 ×
= 396 - Since, the remainder is
and there are no more digits left in the given number, we get: 1296 . - We have found the square root of 1296.
Example 10: Find the least number that must be subtracted from 5607 so as to get a perfect square. Also find the square root of the perfect square.
- Place a bar over every pair of digits starting from the one’s digit. (56 07)
- From the left, find the square which is less than or equal to the digit. Here we have, (
7 2 8 2 when subtracting 7 2 - Bring down the number under the next bar (i.e.076) to the right of the remainder. We now get the new dividend i.e.
- In the divisor put the first digit as double the quotient and place a blank its right. i.e. 14_
- Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
- We get that: 144 ×
= 576 - Since, the remainder is
and quotient is . Thus 74 2 5607. - Thus, the least number that must be subtracted from 5607 to get a perfect square:
.
Example 11: Find the greatest 4-digit number which is a perfect square.
- Let's take the greatest 4-digit number i.e.
- Place a bar over every pair of digits starting from the one’s digit. (99 99)
- From the left, find the square which is less than or equal to the digit. Here we have, (
9 2 10 2 when subtracting 9 2 - Bring down the number under the next bar (i.e. 99) to the right of the remainder. We now get the new dividend i.e.
- In the divisor put the first digit as double the quotient and place a blank its right. i.e. 18_
- Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
- We get that: 189 ×
= 1701 - Since, the remainder is
and there are no more digits left in the given number, we can say that we need to subtract from in order to get the largest four-digit perfect square. Thus, the answer is - We also find that
9801 .
Example 12: Find the least number that must be added to 1300 so as to get a perfect square. Also find the square root of the perfect square.
- Place a bar over every pair of digits starting from the one’s digit. (13 00)
- From the left, find the square which is less than or equal to the digit. Here we have, (
3 2 4 2 when subtracting 3 2 - Bring down the number under the next bar (i.e. 00) to the right of the remainder. We now get the new dividend i.e.
- In the divisor put the first digit as double the quotient and place a blank its right. i.e. 6_
- Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
- We get that: 66 ×
= 396 - Since, the remainder is
and there are no more digits left in the given number. - Thus, we have
as our quotient which tells us that 36 2 37 2 . - Thus, the least number that must be added to 1300 so as to get a perfect square: 1369 - 1300 =
.