Square and Square Roots

Square and Square Roots
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Square Number

Product we get when a number is multiplied with itself is called a square number.

or result we obtain when a number is squared (raised to power of 2) is called a square number.

2×2 = 4, 4/3 × 4/3 = 16/9 etc.

Perfect Square

When we multiply any integer with itself, we always get a whole square number called perfect square. So, a perfect square is a square of an integer.

(-2)×(-2) = 4,

(-1)×(-1) = 1,

1×1 = 1,

2×2 = 4,

3×3 = 9, etc

● For now, we consider Squares of only positive integers i.e, natural numbers,

1×1 = 1,

2×2 = 4,

3×3 = 9, etc

Properties of Square numbers

Number Square Number Square
1

2

3

4

5

6

7

8

9

10

1

4

9

16

25

36

49

64

81

100

11

12

13

14

15

16

17

18

19

20

121

144

169

196

225

256

289

324

361

400

➤ All perfect square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. None of these end with 2, 3, 7 or 8 at unit’s place.

↪ When a square number ends in 1, the number whose square it is, will have either 1 or 9 in unit’s place.

1^2 = 1,\, 11^2 = 121\\  \ 9^2 = 81,\, 19^2 = 361

↪ When a square number ends in 4, the number whose square it is, will have either 2 or 8 in unit’s place.

2^2 = 4,\, 12^2 = 144\\  \ 8^2 = 64,\, 18^2 = 324

↪ When a square number ends in 9, the number whose square it is, will have either 3 or 7 in unit’s place.

3^2 = 9,\, 13^2 = 169\\  \ 7^2 = 49,\, 19^2 = 289

↪ When a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit’s place.

42 = 16,\, 142 = 196\\  \ 62 = 36,\, 162 = 256

↪ When a square number ends in 5, the number whose square it is, will have 5 in unit’s place.

5^2 = 25,\, 15^2 = 225

↪ When a square number ends in 0, the number whose square it is, will have 0 in unit’s place.

10^2 = 100,\, 20^2 = 400

↪ Square numbers can only have even number of zeros at the end which must be double the number of zeros of the number whose square it is.

10^2 =100,\, 20^2 = 400,\\\ 30^2 =900,\, 100^2 = 10000

↪ The squares of even numbers are even numbers and squares of odd numbers are odd numbers.

8^2 =64,\, 16^2 = 256,\\  \ 7^2 =49,\, 15^2 = 225,

Finding the Square of a Number

→ Multiply the number with itself

2^2 = 2\times 2 =4\\  \ 7^2 = 7\times 7 =49

→ Using algebraic identity –

Write the number as sum or difference of two appropriate numbers and then use one of these identities

(a+b)^2 = a^2 +b^2 +2ab,\, or

(a-b)^2= a^2 +b^2 -2ab .

26^2 = (20+6)^2 = 20^2 +6^2 +2\times 20\timed 6\\  \ = 400 + 36 + 240 = 676

or,

262 = (30-4)^2 = 30^2 +4^2 -2\times 30\times 4\\  \ = 900 + 16 - 240 = 916-240 = 676

Patterns in Squares

➤ Adding Triangular numbers

Triangular numbers – Numbers whose dot patterns can be arranged as triangles.

1 3 6 10 15
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If we combine two consecutive triangular number we get a square number,

1+3 = 4

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= 🔴🔵

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3+6 = 9

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= 🔴🔵🔵

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6+10 = 16

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= 🔴🔵🔵🔵

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🔴🔴🔴🔴

Square Roots

↪ Square root of a number is the value which when multiplied by itself gives that number.

↪ Finding the number having the known square is known as finding the square root, it is the inverse operation of squaring.

4 = 2×2, Square Root of 4 is 2.

25 = 5×5, Square Root of 25 is 5.

↪ Every perfect square has two integral square roots, one is positive and other is its negative,

4 = 2×2, also 4 = (-2)×(-2), so square roots of 4 are −2 and 2.

9 = 3×3, also 9 = (-3)×(-3), so square roots of 9 are −3 and 3.

Negative square root of a number is denoted by the symbol -√.

Positive square root of a number is denoted by the symbol √.

e.g., −√4 = −2 & √4 = 2

↪ For now, we consider only positive square roots.

Statement Inference Statement Inference
1^2 = 1 \sqrt1 = 1 6^2 = 36 \sqrt{36} = 6
2^2 = 4 \sqrt4 = 2 7^2 = 49 \sqrt{49} = 7
3^2 = 9 \sqrt9 = 3 8^2 = 64 \sqrt{64} = 8
4^2 = 16 \sqrt {16} = 4 9^2 = 81 \sqrt{81} = 9
5^2 = 25 \sqrt {25} = 5 10^2 = 100 \sqrt{100} = 10

Finding Square root

Prime Factorisation method

→ Write the given number as product of its prime factors

→ Pair the prime factors.

→ Express the paired prime factors in exponential form of power of 2

→ Take the power of 2 as common

→ Put √ before the number in LHS and remove the power in RHS to get the required value.

e.g. 144 = 2×2×2×2×3×3

⇒ 144 = 2×2×2×2×3×3

\implies 144 = 2^2 \times 2^2 \times 3^2\\  \ \implies 144 = (2 \times 2 \times 3)^2\\  \ \implies \sqrt{144} = 2\times 2\times 3 = 12\\  \ \therefore \sqrt{144} = 12

Division method

➧ For large numbers we use long division method.

➧ If a perfect square is of n-digits, then its square root will have n/2 digits for even n or (n +1)/2 digits for odd n.

➧ Steps in long division method

Step 1 ⇾ Make pair of digits in given number starting with digit at one’s place. Put bar on each pair. If the number of digits of the given number is odd, then place the bar on this remaining digit also.

Step 2⇾ Find the divisor, such that its square is either equal or less than the number under first bar from the left. Here divisor and first digit of quotient will be same. Now do the division and get the remainder.

square root

Step 3⇾ Now bring down the next digits under second bar and place them to the right of the remainder. This will become new dividend.

Step 4⇾ Starting digit(s) of new divisor is equal to twice of first digit of quotient (calculated in step 2)

Step 5⇾ Now the digit at one’s place of new divisor is equal to next digit of quotient. And it is to be selected in such a way that, when new divisor is multiplied by this next digit of quotient, the product will be equal or less than the new dividend (as calculated in step 3)

Step 6⇾ Divide and get the new remainder.

This process is to be repeated till all numbers are used under the bars.

Square root of a Decimal

Division method

⇾ To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place. Proceed as usual.

⇾ Put a decimal point in the quotient when the decimal part in the number is taken for the division.

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