8. Square and Square Root Cube and Cube-Root Operations
| 8. Square and Square Root Cube and Cube-Root Operations |
Learn and Remember :
1. When a number is multiplied by itself the product is called the square of that number and that number is called the square root of the product.
For example, 3 x3=9 (9 is a square of 3 and 3 is the square root of 9).
2. The square of a natural number is called a perfect square number.
3. The digits 2, 3, 7 and 8 are not seen in the units place of the perfect squares.
4. A number having the digit 0, 1, 4, 5, 6 or 9 in its units place, is likely to be a perfect square. If 5 exists in the units place of a perfect square, then the digit 2 must exist in its tens place.
e.g. 152 = 225 , 452 = 2025
5. A perfect square ending with zeros, must have an even number of zeros at the end. e.g. 4900 is a perfect square of 70, but 490 is not a perfect square. 90000 is a perfect square of 300, but 9000 is not a perfect square.
6. Square root of a vulgar fraction is equal to the square root of its numerator divided by the square root of its denominator.
7. If a decimal fraction is a complete square, it must have an even number of decimal places
(excluding zeros at the end).
8. Square root of a negative number cannot be evaluated.
9. To find the square of a number with the digit 5 in its units place :
(i) Consider the number excluding 5 in the units place.
(ii) Take the product of that number and the next natural number.
(iii) Write 25 after that product.
e.g. (i) Evaluate (85)2 : 8 x 9 =72 ∴ (85)2 = 7225
(ii) Evaluate (205)2 : 20 x 21 = 420 ∴ (205)2 = 42025.
10. Formulae for squaring :
(1) (a +b)2 = a2+2ab+ b2
Ex. Find the square of 102
102 = 100 +2
(102)2 = (100+ 2)2
= (100)2 +2(100)(2)+ (2)2
= 10000 + 400+ 4 = 10404
∴ (102)2 = 10404
(2) (a-b)2= a2-2ab+b2
Ex. Find the square of 97.
97=(100-3)
(97)2
= 1000 - 600 +9 =9409
∴ (97)2 = 9409
11. Square root : To denote the square root of a number, the symbol ' √ is used.
e.g. square root of 64 is written as √64.
12. To find the square root of a perfect square by factorization :
(1) Write the given number in the form of the product of its prime factors.
(2) Pair the equal factors.
(3) Take one factor from each pair and obtain the product of those factors.
For example, 36 = 2 x 2 x 3 x 3 , ∴ √36 = 2 x3=6.
13. To find orally the square root of a perfect square, in which the digits in the tens and units places form the number 25 :
(1) Consider the number excluding the digits 2 and 5 in tens and units places respectively.
(2) Find two factors of that number which are consecutive numbers.
(3) Take the smaller of the two factors and put the digit 5 after that factor.
e.g. find the square root of 15625. Excluding the last two digits, the number is 156.
156 = 12 x 13.
The smaller factor is 12.
∴ the square root of 15625 is 125, i.e. √15625 = 125.
| The digit at the units place of a square number | 1 | 4 | 9 | 6 | 0 | 5 |
| The digits at the units place of square root | 1 , 9 | 2 , 8 | 3 , 7 | 4 , 6 | 0 | 5 |
[Note : For square numbers : If the units place digit is 0, the tens place digit should also be 0. If the units place digit is 5, the tens place digit must be 2.]
14. To find the square root of a perfect square by the method of approximation :
Let us understand this method by an example.
Example : Find the square root of 7569.
1) Digit in the units place of the number is 9.
∴ 7569 may be a perfect square.
2) (80)2 = 6400 and (90)2 = 8100
6400 < 7569 < 8100. :. 80 < √7569 < 90.
(3) The digit in the units place of 7569 is 9. Hence, the digit in the units place of the
square root is either 3 or 7.
∴ the square root is either 83 or 87.
(4) 7569 is nearer to 8100 than 6400.
∴ the square root should be nearer to 90 than 80.
∴ the square root should be 87.
(5) Confirm the answer.
A short cut : The number obtained by adding 16 to the product of any four consecutive odd natural numbers or even natural numbers is a perfect square. The square root = the product of the extreme numbers + 4. For example, (i) 5 x 7 x 9x 11 + 16 = 3481. √3481 = 5 x 11+4= 59. (ii) 4 x 6 x 8 x 10+ 16 = 1936. √1936 = 4 x 10+4 = 44. • The number obtained by adding 1 to the product of any four consecutive natural number is a perfect square. The square root = the product of extreme numbers +1. For example, √6 x 7 x 8 x 9) + 1 = 3024+ 1 = 2035. √(6 7 x 8 x 9) +1 =√3025 = 6 x 9+1=54 + 1 = 55. |
Cube, Cube root : |
1. Cube : The result of multiplying a number by itself three times is the cube of the number.
e.g. 7 x 7 x 7 = 343
∴ 73 = 343.
Here, the number 7 is multiplied itself three times.
In 73 , 7 is called the base and 3 is called the power.
• The cube of a positive number is positive and that of a negative e number is negative.
• The cubes of natural numbers from 1 to 10 :
| Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Cube | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
2.Cube root : The number whose cube is a given number is called the cube root of the given number. e.g. the cube of 5 is 125.
5 is the cube root of 125.
The cube root of 125 is written as 3√125.
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