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Squares and Cubes

Squares and Square Roots

Square of a number: If a number is multiplied by itself, then the result is called the square of that number. In other words, the square of a number is that number raised to the power 2.

Mathematically, we can say that: if 'm' and 'n' are two numbers such that m * m = n, then 'n' is the 'square of m' and called as 'square number'. For example, in the product 8 * 8 = 64, 64 is the square number.

Example: How many numbers lie between the squares of 10 and 11?
Solution: 102 = 100
112 = 121
The numbers between 121 and 100 are: 101, 102, . . ., 120.
∴ 20 numbers are there between 102 and 112.

Perfect squares: A natural number is called a perfect square, if it is square of some other natural number. Examples: 4 (2 × 2), 25 (5 × 5), 64 (8 × 8), . . .

Note: A given number is a perfect square, if it can be expressed as the product of exact number of pairs of equal factors.

If there are 'n' digits in given number, the square will have either 2n or 2n – 1 digits. This condition is true only for numbers above '3', because the square of 2 = 4 and the square of 3 = 9.

For example, 52 = 25. Here '5' is a single digit number (n = 1) whereas 25 is two digits number (2n = 2 × 1 = 2). Similarly, 122 = 144. Here '12' is a two digits number (n = 2) where as 144 is three digits number (2n – 1 = 2 × 2 – 1 = 3).

Finding the Square of a Number

There are several methods to find the square of a number in Mathematics. The method we shall discuss in this topic gives a general procedure of finding square of any number and is based on a process known as Duplex Combination Process.

The term "Duplex" is used in two different senses. The first one is by squaring and the second one is by cross-multiplication. In the present context, it is used in both the senses (a2 and 2ab).

Duplex of a number
• For a single-digit number, duplex is its square.
Ex: Duplex of 8 is: 82 = 64.
• For a two-digit number, duplex is twice the product of the two digits.
Ex: Duplex of 36 is: 2(3 × 6) = 2 × 18 = 36.
• For a three-digit number, duplex is twice the product of the outer two digits plus the square of the middle digit.
Ex: Duplex of 745 is: 2(7 × 5) + 42 = 70 + 16 = 86.
• For a four-digit number, duplex is the sum of two products: first, twice the product of the extreme digits, and second, twice the product of the middle two digits.
Ex: Duplex of 4723 is: 2(4 × 3) + 2(7 × 2) = 24 + 28 = 52.
In the similar manner, we can write duplex for n-digit number.

The process of finding the square of a no. is best illustrated with the help of .

Squaring of numbers ending in 5:
Step 1: Square 5 and put down 25 as the right hand part of the answer.
Step 2: After dropping 5, multiply the number left by a number 1 more than itself. This gives left hand part of the answer.

Points to remember

1. The sum of squares of an odd number and an even number is an odd number.

eg: 132 + 142 = 169 + 196 = 365 (odd).

2. The sum of squares of two even numbers is an even number.

eg: 162 + 122 = 256 + 144 = 400 (even).

3. The sum of squares of two odd numbers is an even number.

eg: 172 + 112 = 289 + 121 = 410 (even).

4. The product of the squares of odd number and even number is an even number.

eg: 22 * 32 = 4 * 9 = 36 (even).

5. The product of the squares of two even numbers is an even number.

eg: 22 * 42 = 4 * 16 = 64 (even).

6. The product of the squares of two odd numbers is an odd number.

eg: 52 * 32 = 25 * 9 = 225 (odd).

7. For any natural number 'n' we have (n + 1)2 – n2 = (n + 1) + n.

eg: 92 – 82 = 9 + 8 = 17.

8. A triplet (a, b, c) of three natural numbers a, b and c is called a Pythagorean triplet if a2 + b2 = c2.

9. The square of natural number 'n' is equal to the sum of first 'n' odd numbers.

eg: 1 + 3 = 22, 1 + 3 + 5 = 32, 1 + 3 + 5 + 7 = 42 .....

10. There are no natural numbers 'm' and 'n' for which mn = 2n2.

11. For any natural numbers 'a' and 'b' we have √ab = √a × √b and .

12. If a number M is a perfect square then the next immediate square is M + 2√M + 1.

eg: Take M = 4, a square number.
Now, M + 2√M + 1 = 4 + 2√4 + 1 = 4 + 4 + 1 = 9.

13. If a square number ends in 6 the preceding figure is an odd number.

eg: 62 = 36, 162 = 256, 242 = 576, ..............., etc.

14. For every natural number 'n', (n + 1)2 – n2 = 2n + 1. Hence every odd number (2n + 1) can be expressed as difference of two consecutive perfect squares.

eg: (1 + 1)2 – (1)2 = 22 – 12 = 4 – 1 = 3.

15. There are no natural numbers p and q such that p2 = 2q2.

16. If n is a perfect square then '2n' can never be a perfect square.

eg: 4 and 9 are perfect squares, but 2 × 4 and 2 × 9 are not perfect squares.

17. For any natural number 'n' greater than 1 the Pythagorean triplet is given by (2n, n2 – 1, n2 + 1)

eg: n = 6 then 2 × 6, 62 – 1, 62 + 1 represents a Pythagorean triplet. i.e. 12, 35, 37 because 122 + 352 = 372.

18. We can express the square of any odd number as the sum of two consecutive positive integers.

eg: 52 = 25 = 12 + 13, 72 = 49 = 24 + 25, ..............., etc.

19. The converse of the above statement is not true. i.e. the sum of any two consecutive positive integers is not necessarily a perfect square.

eg: 14 + 15 = 29, which is not a perfect square.

Square Root

Square root of a number:
The square root of a number 'n' is that number which when multiplied by itself gives 'n' as the product. The symbol used for square root is . We denote the square root of the number 'n' by √n.

We know that 22 = 4, 32 = 9, 122 = 144. Then, 2 is called square root of 4, 3 is called square root of 9 and 12 is called square root of 144. Thus, if m2 = n, then 'm' is called square root of 'n'.

Principal square root

(5)2 = 25 and also (– 5)2 = 25.
So, – 5 is also a square root of 25. In other words, 25 has two square roots which are 5 and – 5. It is denoted as (± 5).
But the principal square root of a number is the positive square root only. Hereafter, when we say "the square root" it implies we are referring to the principal(or positive) square root unless specifically mentioned other wise.

Properties of square roots
Based on properties of square numbers, we have some properties on square roots.
1. If the units digit of a number is 2, 3, 7 or 8, then it is not a perfect square and hence does not have a square root.
2. If a number has a square root, then its units digit must be 0, 1, 4, 5, 6 or 9.

Units digit of square 0    1    4 5    6    9
Units digit of square root 0 1 or 9 2 or 8 5 4 or 6 3 or 7
3. If a number ends in an odd number of zeros, then it does not have a square root.
Example: 1000, 20, 90, 150.
4. The square root of an even square number is even and the square root of an odd square number is odd.
Prime factorization method for finding square roots
When a given number is a perfect square, we define its square root as given below:
Step 1: Resolve the given number into prime factors.
Step 2: Make pairs of equal primes.
Step 3: The product of prime factors, chosen one out of every pair, gives the square root of the given number.
  • Example : Find the smallest number by which 2400 should be multiplied to get a perfect square number ?
  • Sol: Resolving 2400 into prime factors, we get

    Square root of 2400 by prime factorisation

      2400 = 2 × 2 × 2 × 2 × 5 × 5 × 2 × 3
              = ( 2 × 2) × ( 2 × 2) × ( 5 ×5) × ( 2 × 3 )

    Clearly, in the product of prime factors, there are 3 pairs of equal factors and two factors 2 and 3 which do not exist in pairs.

    So, we should multiply the given number by 2 × 3 = 6 to get a perfect square number.

    ∴   Perfect square number so obtained = 2400 × 6
                                                                = 14400 .

    ∴     14400  =   2 × 2 × 5 × 2 × 3   =   120


Square roots by division method: The method of finding square roots by the prime factorization method is efficient only if the number has small prime factors. Sometimes it is difficult and time consuming to obtain prime factors of a given number. To over come this difficulty, we use an alternative method called the 'division method'.

Square roots by division method
Step 1: Place a bar over every pair of digits starting from the units digit.
Step 2: Find the largest number whose square is less than or equal to the number under the left–most bar. Take the number as divisor and the number under the left most bar as the dividend. Divide and get the remainder.
Step 3: Bring down the number under the next bar to the right of the remainder. This is the new dividend.
Step 4: Double the quotient and enter it with a blank on the right for the next digit of the next possible divisor.
Step 5: Guess a largest possible digit to fill the blank and also to become the new digit in the quotient.

Square root of a decimal number
We may find the square root of a decimal number without converting into a rational number. We do it as follows:
Step 1: Place bars on the integral part of the number in the usual manner.
Step 2: Place bar on the decimal part on every pair of digits beginning with the first decimal place.
Step 3: Start finding the square root by the division process as usual.
Step 4: Place decimal point in the quotient as soon as the integral part is exhausted.
Step 5: Stop when the remainder becomes zero. The quotient at this stage is the square root.

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