Monday, 28 October 2013
Thursday, 10 October 2013
Cube and Cube Root( as per PSEB )
Cube and Cube Root
Points to Remember:
1.
If a number is multiplied by itself thrice
then the product obtained is cube of the number. Let a number be ‘a’ then cube
of a = a x a x a which is written as a3 and read as cube of a.
2.
1, 8, 27 ….. 1000 are perfect cubes.
3.
Cube of an odd number is odd.
4.
Cube of an even number is even.
5.
Cube of positive number is positive.
6.
Cube of negative number is negative.
7.
If unit’s place digits of a number is 1, 4,
5, 6, 9 and 0 then unit’s place digit of its cube is 1, 4, 5, 6, 9 and 0
respectively.
8.
If unit’s place digit of a number is 2 then
unit’s place digit of its cube is 8.
9.
If unit’s place digit of a number is 8 then
unit’s place digit of its cube is 2.
10. If
unit’s place digit of a number is 3 then unit’s place digit of its
cube is 7.
11. If
unit’s place digit of a number is 7 then unit’s place digit of its
cube is 3.
Points to Remember:
1. If
the cube of a number m = m3 = n then cube root of n is m and we will write 3Ön = m.
2. To
find the cube of a number, we add 1+ (n) (n – 1) x 3 in the previous cube ;
when n = 1, 2, 3 …… etc.
e.g. for n = 1, 1 + (1) (1 – 1) x 3 =1
for n = 2, 1 + (2) (2 – 1) x 3 =7
for n = 3, 1 + (3) (3 – 1) x 3 =19
for n = 4, 1 + (4) (4 – 1) x 3 =37
(
………………………………….)
For n = 9, 1 + (9) (9 - 1) x 3 = 217
Thus
by adding 1, 7, 19, 37, 61 ………… 217 etc, to the previous cube, the next cube is
obtained.
3. If by subtracting above numbers i.e. 1, 7,
19, 37, 61, 91, 169, 217,
271 etc. in order from the given number,
the remainder is zero, then
the number is perfect cube otherwise it
is not a perfect cube.
Points to Remember:
1.
The
numbers having unit’s place digit
(i) 0, 1, 4, 5, 6 and 9 then their cubes have
unit’s place digit as 0,
1, 4, 5, 6 and respectively.
(ii) 2, then unit’s place digit of its cube is 8
(iii)
8, then unit’s place digit of its cube is 2
(iv)
3, then unit’s place digit of its cube is 7
(v) 7, then unit’s place digit of its cube is 3
Therefore, the unit’s place digit of
cube root of a number is known by the unit’s place digit of the number.
2. (i) Cube of one digit
number has maximum three digits.
(ii) Cube of two
digit number has maximum six digits.
Points to Remember:
1. Cube of a negative number is negative and cube root of a
negative number is also negative 3Ö-m=-3Öm
2. For
a rational number p, 3Öp =3Öp
q
q Öq
3. If
there are two numbers a and b then 3Ö a x b = 3Ö a x 3Öb
Square and Square Root ( as per PSEB )
Square and Square Root
Points to Remember:
1.
When a number is multiplied be itself then
the product is the square of the number.
2.
Square of an odd number is odd.
3.
Square of an even number is even.
4.
Perfect square has only 0, 1, 4, 5, 6 or 9
digits at its unit’s place.
5.
If a number has its unit’s place digits 2, 3,
7 or 8 then it is not a perfect square.
6.
If a number has its unit’s place digits 1 0r
9 then its square has 1 as unit’s place digit.
7.
If
unit’s place digit of a number is 2 or 8 then its square has 4 as unit’s place
digit.
8.
If unit’s place digit of a number is 3 or 7
then its square has 9 as unit’s place digit.
9.
If unit’s place digit of a number is 4 or 6
then its square has unit’s place digit 6.
10. If
unit’s place digit of a number is 5 then its square has unit’s place digit 5
and ten’s place digit 2 i.e. the square ends with 25.
11. If
unit’s place digit of a number is 0 (zero) then its square has
unit’s and ten’s place digit as zero.
12. When
a perfect square is divided by 3 remainder is 0 or 1.
Square and Square Root
Points to Remember:
1.
By multiplying a number by itself we get the
square of the number.
2.
Square of positive and negative numbers is
always a positive number and no negative number is a perfect square.
3.
By using the identities (a+b)2 = a2+
2ab + b2 and (a – b) 2 = a2 – 2ab + b2
we can find the square of numbers.
Points to Remember:
1.
If n= m2 then n is square root of m and we
write Ön = m.
2.
If we go on subtracting 1, 3, 5, …… in order,
at one stage the remainder will be zero. If the number of subtractions is n
(say n times) then he square root of this number is n.
Points to Remember:
1.
Square root by prime-factorization method:
(i)
Prime factors of number are made.
(ii)
Pairs of equal prime factors of number are made.
(iii)
One factor out of a pair of equal factors is taken.
(iv)
By multiplying the one-one factor taken from pair of equal
factors square root of the number is
obtained.
2. If the prime factors are not in pairs
then number is not a perfect
square because perfect square is formed
by multiplying two equal
numbers.
We know that if equal prime factors are not in pairs then
number is not a perfect square but this can be made another number by
multiplying or dividing it by some prime factors which is a perfect square.
To find the number of digits in the square root of a
number, starting from unit place put dot (·) on
the alternate digits of the number. The number of digits in square root is the
same as the number of dots.
Points to Remember:
1. Rational number: A
number which can be put in the form of p/q
Where p, q are integers, q ≠ 0 (i.e. q is
not zero), p and q having
no common factor is called rational number.
2. For taking square
root of rational number we write Öp = Öp
q
Öq
Points to Remember:
1.
To find the square root of decimal numbers
(a)
When these numbers are perfect squares:
(i)
Starting from decimal on right and left side make the pairs.
(ii) Start doing square root as usual.
(iii) Whenever turn of decimals comes, put decimal
in quotient.
(iv) Proceed till remainder is zero. The obtained
quotient is square root of the number.
(b) When numbers are not perfect
squares: In such questions to find out square root upto certain decimal places
we put zero after the decimal point to complete the pairs because by putting zero
after the decimal the value of number does not change.