The document discusses rational numbers and their properties with respect to addition and multiplication. It also covers exponents and laws regarding powers and quotients of powers. Rational numbers can be added, multiplied, and follow properties like closure, commutativity, associativity, and inverses. Exponents represent the power to which a base is raised. Laws state that in a product of powers, exponents are added, powers of products are equal to the product of powers, and quotients of powers involve subtracting the bottom exponents from the top exponent.

1. CHAPTER : 2
SYSTEM OF REAL NUMBER, EXPONENTS & RADICALS
RATIONAL NUMBERS:
The fraction written in the form of “P/q” (where p, q E Z and q ≠ 0) is called
rational number.
PROPERTIES OF RATIONAL NUMBERS:
The rational number shows the properties with respect to addition and
multiplication.
PROPERTIES WITH RESPECT TO ADDITION:
We have three number (Rational) a, b and c then.
1. a + b => result is also a rational No. (Closure property).
2. a + b = b + a => Result is same on b. sides. (Commutative property)
3. a + (b + c) = (a + b) + c => Result in same on b. sides.(Associative property)
4. a + 0 = 0 + a => Identities.
5. a + (-a) = 0 = (-a) + a => Result is zero.(Inverse property)
6. a(b + c) = ab + ac => (Distributive property)
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2. PROPERTIES WITH RESPECT TO MULTIPLICATION:
We have three numbers (rational) a, b, c then.
1. a b => Result is also a Rational number.(closure
property)
2. a. b = b. a => Result is same on b.sides.(commutative
property)
3. a (bc) = (ab) c => Result is same on b.
sides.(Associative property)
4. a x 1 = a = 1 x a => Identities.
5. a x = 1 = x a => Result is one.(Inverse property)
6. (b+c)a = ba + ca => (Distributive property)
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3. EXPONENT:
The nth power of a number “a” is called base and “n” the exponent.
Base a n Exponent
(i) In the product of different powers having the same base, the exponents are
added but the base is remains uncharged.
i.e: am x an = am + n
(ii) If we have a numbers contain two numbers then power product of two
number is equal to the product of their power.
(a . b)n = a n . b n
REMEMBERED POINT:
(i) If “a” is a positive real number, then an is positive.
Example: (5)3 = 125, (0.5)3 = 0.125, (0.5)2 = 0.25
(ii) If “a” is a negative real number and “n” is even, then an is positive.
Example: (-5)2 = 25, (-0.5)2 = 0.25, (-5)4 = 625.
(iii) If “a” is a negative real number and “n” is odd then an is negative.
Example: (-5)3 = -125, (-0.5)3 = 0.125, (-2)5 = -32
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4. LAW OF POWER OF A POWER:
In the case of power of a power, the base remains the same while we get a new
exponent of the base which is equal to the product of the two exponents.
•(am ) n = am x n
•[(a m) n] p = (am n) p = am n p
LAW OF QUOTIENT OF POWER:
In the quotient powers on the same base we subtract the power of the
denominators from that number (Numerator) and the base remains
unchanged.
* am / an = a m - n
* (a / b)-n = a-n/b-n = b n /an
* (a / b) n = a n/b n
* a- n = 1 /an
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