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A square root is a value that, when multiplied by itself, gives the original number.

We can use two methods to find square roots - Prime factorization and the Long division method.

Just the opposite method of squaring a number is the square root. If a number n, such as n^{2}, is squared, then the square root of n^{2} is equal to the original number n.

We need to multiply the number by itself to find the square of a number.

For example, 3 multiplied by 3 is equal to 9.

Similarly,

Square of 4 is: 4 multiplied by 4 = 4 × 4 = 16.

Q1: Can you find out the number of numbers lying between the squares of these following pairs of numbers?

(i) 25 and 26

(ii) 99 and 100

Solution - As we know, between n^{2} and (n + 1)^{2}, the number of non–perfect square numbers are 2n.

(i) Between 252 and 262 there are 2 × 25 = 50 natural numbers.

(ii) Between 992 and 1002 there are 2 × 99 = 198 natural numbers.

Q2: Write a Pythagorean triplet whose one of the required member is:

(i) 6

(ii) 14

(iii) 16

(iv) 18

Solution -

We know, for any natural number m, 2m, m^{2} – 1, m^{2} + 1 is a Pythagorean triplet.

(i) 2m = 6

⇒ m = 6/2 = 3

m^{2} – 1 = 3^{2} – 1 = 9 – 1 = 8

m^{2} + 1 = 3^{2} + 1 = 9 + 1 = 10

Therefore, (6, 8, 10) is a Pythagorean triplet.

(ii) 2m = 14

⇒ m = 14/2 = 7

m^{2} – 1 = 72 – 1 = 49 – 1 = 48

m^{2} + 1 = 72 + 1 = 49 + 1 = 50

(14, 48, 50) is not a Pythagorean triplet.

(iii) 2m = 16

⇒ m = 16/2 = 8

m^{2} – 1 = 8^{2} – 1 = 64 – 1 = 63

m^{2} + 1 = 8^{2} + 1 = 64 + 1 = 65

Therefore, (16, 63, 65) is a Pythagorean triplet.

(iv) 2m = 18

⇒ m = 18/2 = 9

m^{2} – 1 = 92 – 1 = 81 – 1 = 80

m^{2} + 1 = 92 + 1 = 81 + 1 = 82

Therefore, (18, 80, 82) is a Pythagorean triplet.

Q3: (n + 1)^{2} - n^{2} = ?

Solution -

(n + 1)^{2} - n^{2}

= (n^{2} + 2n + 1) – n^{2}

= 2n + 1

Q4: State that the number 121 is the sum of 11 odd natural numbers.

Solution - As 121 = 112

We recognize that n2 is the sum of the first n odd natural numbers.

It shows that 121 = the sum of the first 11 odd natural numbers.

= 1 + 3 + 5+ 7 + 9 + 11 +13 + 15 + 17 + 19 + 21

Q5: Use the identity and find the square of 189.

(a – b)^{2} = a^{2} – 2ab + b^{2}

Solution - 189 = (200 – 11)^{2}

= 40000 – 2 × 200 × 11 + 112

= 40000 – 4400 + 121

= 35721

Q6: Find out the square root of 625 using the mathematical identity as stated: (a + b)^{2} = a^{2} + b^{2} + 2ab?

Solution - (625)^{2}

= (600 + 25)^{2}

= 6002 + 2 × 600 × 25 + 252

= 360000 + 30000 + 625

= 390625

A square root function is defined in mathematics as a one-to-one function that takes as an input a positive number and returns the square root of the given input number.

f(x) = √x

The following are some of the essential properties of the square root:

If a number is a perfect square number, a perfect square root exists.

It may have a square root if a number ends with an even number of zeros (0's).

It is possible to multiply the two square root values. For instance, √3 can be multiplied by √2, then √6 should be the result.

If two same square roots are multiplied, then a radical number should be the product. This implies that a non-square root number is a product. When √7 is multiplied by √7, for example, the result obtained is 7.

It does not define the square root of any negative numbers. And no negative can be the perfect square.

If a number ends with 2, 3, 7, or 8 (in the digit of the unit), the perfect square root will not exist.

If in the unit digit, a number ends with 1, 4, 5, 6, or 9, the number would have a square root.

Using the prime factorization process, the square root of a perfect square number is simple to measure. For example -

With the help of an example, let us understand this concept:

Example 1: Solve √10 to 2 decimal places.

Solution -

Step 1: Choose any two perfect square roots between which you feel your number may fall.

We know that 22 = 4; 32 = 9, 42 = 16 and 52 = 25

Now, choose 3 and 4 (as √10 lies between these 2 numbers)

Step 2: Divide the given number into one of the square roots chosen.

Divide 10 by 3.

=> 10/3 = 3.33 (round off answer at 2 places)

Step 3: Find the root average and the product of the step above, i.e.

(3 + 3.33)/2 = 3.1667

Verify: 3.1667 × 3.1667 = 10.0279 (Not required)

Repeat step 2 and step 3

Now 10/3.1667 = 3.1579

Average of 3.1667 and 3.1579.

(3.1667+3.1579)/2 = 3.1623

Verify: 3.1623 × 3.1623 = 10.0001 (more accurate)

Stop the process.

Example 2: Find the square roots of whole numbers from 1 to 100 that are perfect squares.

Solution - The perfect squares are - 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Example 3: What is:

The square root of 2.

The square root of 3.

The square root of 4.

The square root of 5

Solution - Use a square root list, we have

Value of root 2 i.e. √2 = 1.4142.

Value of root 3 i.e. √3 = 1.7321.

Value of root 4 i.e. √4 = 2.

Value of root 5 i.e. √5 = 2.2361.

FAQ (Frequently Asked Questions)

Question 1. Define Square Numbers and Square Roots.

Answer: Square numbers can be defined as the numbers that are produced when a number is multiplied by itself. For example, if n is a number and it is multiplied by itself, then the square of n can be given as n^{2}. Another example is the square of 10, which is 10^{2} = 10 x 10 = 100.

Further, the square root of a number can be explained as the value which when multiplied by itself gives the original number. This value can be represented by the symbol ‘√.’ For example, the square root of 25 is √25 = 5.

Question 2. What do you Understand by Perfect Squares? Provide Examples with Your Answer.

Answer: Perfect squares can be defined as numbers in which the square root of the number provides a whole number. For example, 9 is a perfect square number because its root is a whole number, which means that √9 = 3.

Question 3. What is an Imperfect Square? Provide Examples with Your Answer.

Answer: An imperfect square can be defined as a number in which the square root of the number is a fraction. The value is generated by taking the square root of the imperfect square can also be non-terminating. For example, 3 is an imperfect square because the root of this number is 1.73205080757. This number is a fraction.

Question 4. What is the Main Difference Between a Square and Square Roots?

Answer: The square root of a number provides the root of a particular number that was squared. This is the main difference between a square number and square roots.

Question 5. How to Solve the Square Root Equation.

Answer: We need to follow the below steps to solve the square root equation: Isolate the square to one of the sides (L.H.S or R.H.S). Now solve the remaining equation. Square both sides of the given equation. With examples, let us understand the steps.

Question 6. Is the Square Root of a Negative Number a Whole Number?

Answer: No, negative numbers shouldn't have a square root, as per the square root concept. If we multiply two negative numbers, a positive number will always be the product. Negative number square roots expressed as multiples of I (imaginary numbers).