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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 â€˜âˆšnâ€™, where â€˜âˆšâ€™ 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, (âˆš3)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.

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.

Letâ€™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, â€˜âˆšâ€™. If the square root of n is a, then a multiplied by a is equal to n. It can be expressed as:

âˆšn = 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 (âˆš) 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:

• âˆš2 = 1.4142
• âˆš3 = 1.7321
• âˆš8 = 2.8284

Square Roots 1 to 50

 Number Square Root Number Square Root Number Square root âˆš1 1 âˆš18 4.2426 âˆš35 5.9161 âˆš2 1.4142 âˆš19 4.3589 âˆš36 6 âˆš3 1.7321 âˆš20 4.4721 âˆš37 6.0828 âˆš4 2 âˆš21 4.5826 âˆš38 6.1644 âˆš5 2.2361 âˆš22 4.6904 âˆš39 6.2450 âˆš6 2.4495 âˆš23 4.7958 âˆš40 6.3246 âˆš7 2.6458 âˆš24 4.8990 âˆš41 6.4031 âˆš8 2.8284 âˆš25 5 âˆš42 6.4807 âˆš9 3 âˆš26 5.0990 âˆš43 6.5574 âˆš10 3.1623 âˆš27 5.1962 âˆš44 6.6332 âˆš11 3.3166 âˆš28 5.2915 âˆš45 6.7082 âˆš12 3.4641 âˆš29 5.3852 âˆš46 6.7823 âˆš13 3.6056 âˆš30 5.4772 âˆš47 6.8557 âˆš14 3.7417 âˆš31 5.5678 âˆš48 6.9282 âˆš15 3.8730 âˆš32 5.6569 âˆš49 7 âˆš16 4 âˆš33 5.7446 âˆš50 7.0711 âˆš17 4.1231 âˆš34 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 â€˜âˆšâ€™. For example, the square root of 25 is âˆš25 = 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.âˆš9 = 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,

âˆš3 = 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:

âˆš900 = âˆš(2 x 2 x 3 x 3 x 5 x 5 )

Taking out the numbers in pairs, we get;

âˆš900 = 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.