Squares and Square Roots

Squares and square roots both concepts are opposite in nature to each other. Squares are the numbers, generated after multiplying a value by itself. Whereas square root of a number is value which on getting multiplied by itself gives the original value. Hence, both are vice-versa methods. For example, the square of 2 is 4 and the square root of 4 is 2.

If n is a number then its square is represented by n raised to the power 2, i.e., n2 and its square root is expressed as ‘√n’, where ‘√’ is called radical. The value under the root symbol is said to be radicand.

The square numbers are widely explained in terms of area of a square shape. The shape of a square is such that it has all its sides equal. Therefore, area of square is equal to (side x side) or side2. Hence, if the side length of the square is 3cm then its area is 32= 9 sq.cm.

Properties of Square Numbers

The square numbers are the values which are produced when we multiply a number by itself. Some of the properties are:

  • Square of 1 is equal to 1
  • Square of positive numbers are positive in nature
  • Square of negative numbers is also positive in nature. For example, (-3)2 = 9
  • Square of zero is zero
  • Square of root of a number is equal to the value under the root. For example, (√3)2 = 3
  • The unit place of square of any even number will have an even number only.
  • If a number has 1 or 9 in the unit’s place, then its square ends in 1.
  • If a number has 4 or 6 in the unit’s place, then its square ends in 6.

Also, read:

Square Numbers 1 to 50

12 = 1

112 = 121

212 = 441

312 = 961

412 = 1681

22 = 4

122 = 144

222 = 484

322 = 1024

422 = 1764

32 = 9

132 = 169

232 = 529

332 = 1089

432 = 1849

42 = 16

142 = 196

242 = 576

342 = 1156

442 = 1936

52 = 25

152 = 225

252 = 625

352 = 1225

452 = 2025

62 = 36

162 = 256

262 = 676

362 = 1296

462 = 2116

72 = 49

172 = 289

272 = 729

372 = 1369

472 = 2209

82 = 64

182 = 324

282 = 784

382 = 1444

482 = 2304

92 = 81

192 = 361

292 = 841

392 = 1521

492 = 2401

102 = 100

202 = 400

302 = 900

402 = 1600

502 = 2500

Squares of Negative Numbers

The squares of negative numbers give a positive value, because if we multiply two negative numbers then it will result in a positive number.

Remember that: (-) x (-) = (+)

Therefore, square of (-n), (-n)2 = (-n) x (-n) = n2

Where n is a number.

Examples:

  • (-5)2 = (-5) x (-5) = 25
  • (-7)2 = (-7) x (-7) = 49

Numbers between Squares

Suppose there are two square numbers n2 and (n+1)2, then total numbers between these two squares are given by 2n.

Let’s say 32 and 42 are two squares.

32 = 9 and 42 = 16

We need to find the numbers present between 9 and 16.

Here, n = 3

Therefore, total numbers between 9 and 16 = 2n = 2 x 3 = 16

Is that correct? Let us check.

9, 10, 11, 12, 13, 14, 15, 16.

As we can see, the numbers between 9 and 19 are 6. Therefore, the formula given above is applicable to all the squares.

Numbers Between n2 and (n+1)2 = 2n, where n is any natural number

Square Roots of Number

As we have already discussed, the square root of any number is the value which when multiplied by itself gives the original number. It is denoted by the symbol, ‘√’. If the square root of n is a, then a multiplied by a is equal to n. It can be expressed as:

√n = a then a x a = n

This is the formula for square root.

Square Roots of Perfect Squares

The perfect squares are the one whose square root gives a whole number. For example, 4 is a perfect square because when we take the square root of 4, it is equal to 2, which is a whole number. Let us see some of the perfect squares and their square roots.

Perfect Squares

Square Root (√)

0

0

1

1

4

2

9

3

16

4

25

5

Square Root of Imperfect Squares

Finding the square root of perfect squares is easy but to find the root of imperfect squares is difficult. The root of the perfect square can be estimated using the prime factorisation method.

The square root of imperfect squares is usually fractions. For example, 2 is an imperfect square because 2 cannot be prime factorised and its square root gives a fractional value.

Examples are:

  • √2 = 1.4142
  • √3 = 1.7321
  • √8 = 2.8284

Square Roots 1 to 50

Number

Square Root

Number

Square Root

Number

Square root

√1

1

√18

4.2426

√35

5.9161

√2

1.4142

√19

4.3589

√36

6

√3

1.7321

√20

4.4721

√37

6.0828

√4

2

√21

4.5826

√38

6.1644

√5

2.2361

√22

4.6904

√39

6.2450

√6

2.4495

√23

4.7958

√40

6.3246

√7

2.6458

√24

4.8990

√41

6.4031

√8

2.8284

√25

5

√42

6.4807

√9

3

√26

5.0990

√43

6.5574

√10

3.1623

√27

5.1962

√44

6.6332

√11

3.3166

√28

5.2915

√45

6.7082

√12

3.4641

√29

5.3852

√46

6.7823

√13

3.6056

√30

5.4772

√47

6.8557

√14

3.7417

√31

5.5678

√48

6.9282

√15

3.8730

√32

5.6569

√49

7

√16

4

√33

5.7446

√50

7.0711

√17

4.1231

√34

5.8310

Frequently Asked Questions on Squares and square roots

What are squares and square roots?

Square numbers are the numbers which are produced when a value is multiplied by itself. Say if n is a number and is multiplied by itself, then the square of n is given by n2. For example, the square of 10 is 102 = 10 x 10 = 100.

The square root of a number is a value which on multiplied by itself gives the original number. It is represented by the symbol ‘√’. For example, the square root of 25 is √25 = 5.

What are perfect squares? Give an example

Perfect squares are the numbers, the square root of which gives a whole number. For example, 9 is a perfect square, because its root is a whole number, i.e.√9 = 3.

What is an imperfect square with examples?

An imperfect square is a number, the square root of which gives a fraction. The value generated by taking the square root of the imperfect square could be non-terminating as well.

For example, 3 is an imperfect square, because its root is equal to,

√3 = 1.73205080757, which is a fraction.

How to find the square root?

To find the square root of a number we can use the prime factorisation method. For example, the square root of 900 is:

√900 = √(2 x 2 x 3 x 3 x 5 x 5 )

Taking out the numbers in pairs, we get;

√900 = 2 x 3 x 5 = 30

900 was a perfect square, but to find the root of an imperfect square, we have to use long division method.

What is the difference between a square and square roots?

The square root of a number gives the root of a number which was squared. This is the primary difference between them.

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