When a number or integer (not a fraction) is multiplied by itself, the resultant is called a ‘Square Number’.
The following are the properties of the square numbers:
When you square a number, you are multiplying it by itself, e.g. 6*6 = 36. When you take the squareroot of a number, you are undoing the square, going backwards from the result of squaring to the input that was originally squared: √ 36 = 6. Similarly, 8*8 = 64, so √ 64 = 8. As long as all the numbers are positive, everything is straightforward.
The process of finding the square root is called evaluation. The square root of a number is denoted by the symbol called the radical sign(√ ).
(i) By the method of Prime Factors:
When a given number is a perfect square, we resolve it into prime factors and take the product of prime factors, choosing one out of every two.
To find the square root of a perfect square by using the prime factorization method when a given number is a perfect square:
Step I: Resolve the given number into prime factors.
Step II: Make pairs of similar factors.
Step III: Take the product of prime factors, choosing one factor out of every pair.
Examples on square root of a perfect square by using the prime factorization method:
1. Find the square root of 484 by prime factorization method.
Solution:
Resolving 484 as the product of primes, we get
484 = 2 × 2 × 11 × 11
√484 = √(2 × 2 × 11 × 11)
= 2 × 11
Therefore, √484 = 22
2. Find the square root of 324.
Solution: The square root of 324 by prime factorization, we get
324 = 2 × 2 × 3 × 3 × 3 × 3
√324 = √(2 × 2 × 3 × 3 × 3 × 3)
= 2 × 3 × 3
Therefore, √324 = 18
(ii) By the method of Long Division:
This method can be used when the number is large and the factors cannot be determined easily. This method can also be used when we want to add a least number or to subtract a least number from a given number so that the resulting number may give a perfect square of some number.
Steps of Long Division Method for Finding Square Roots:
Step I: Group the digits in pairs, starting with the digit in the units place. Each pair and the remaining digit (if any) is called a period.
Step II: Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor and also as the quotient.
Step III: Subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.
Step IV: Now, the new divisor is obtained by taking two times the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.
Step V: Repeat steps (2), (3) and (4) till all the periods have been taken up. Now, the quotient so obtained is the required square root of the given number.
Examples on square root of a perfect square by using the long division method
1. Find the square root of 784 by the longdivision method.
Solution:
Marking periods and using the longdivision method,
Therefore, √784 = 28
2. Evaluate √5329 using the longdivision method.
Solution: Marking periods and using the longdivision method,
Therefore, √5329 =73
If x^{2} = y, we say that the square root of y is x and we write y = x.
Thus, √4 = 2, √9 = 3, √196 = 14.
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3. Number ending in 8 can never be a perfect square.
4. Remember the squares and cubes of 2 to 10. This will help in easily solving the problems.
Q.1. Simplify the following expression.
Answer:
Q.2. Simplify the following expression.
Answer:
Q.3. Assume x to be positive. Multiply the eighth power of the fourth root of x by the fourth power of the eighth root of x. What is the product?
Answer:
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1. What are square numbers and how are they related to square roots? 
2. How can square numbers be classified into odd and even numbers? 
3. What are some properties of square numbers? 
4. How can one find the square root of an integer? 
5. What are some important points to remember when dealing with squares and square roots? 

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