Now we are going to study about Exponents of real numbers and Rationalisation.
So first come to Exponents of real numbers-
It is nothing but the study of powers of rational numbers and the laws obeyed by itself.
Integral Exponents of a Real Number:-
POSITIVE INTEGRAL POWER-
For any real number 'a' and a positive integer n, we define a power n as
a power n = a×a×a×......×a (n factors)
a power n is called the nth power of a. The real number a is called the base and n is called the exponent (index) of the nth power of a.
For example-
2³ = 2×2×2 = 8
NEGATIVE INTEGRAL POWER:-
For any non-zero real number 'a' and a positive integer n, we define
a power -n = 1/a power n
For example-
5-³ = 1/5³ = 1/5×5×5 = 1/125
Laws of Integral Exponents
First Law:- If a is any real number and m, n are positive integers, then a power m × a power n = a power (m+n).
For example-
5⁴ × 5³ = 5⁴+³ = 5 power 7
Second Law:- If a is a non-zero real number and m, n are positive integers, then a power m/a power n = a power (m-n).
For example-
3 power 5 ÷ 3² = 3 power (5-2) = 3³.
Third Law:- If a is any real number and m, n are positive integers, then (apower m) power n = a power m×n = (a power n) power m.
For example-
(3²) power 5 = 3 power 2×5 = 3 power 10
Fourth Law:- If a, b are real numbers and m, n are positive integers, then
1. (ab) power n = a power n × b power n
2. (a/b) power n = a power n/b power n, where b is not equal to 0.
For example-
1. 6⁴ = (2×3)⁴ = 2⁴ × 3⁴
Rational Exponents of a real number
Principal nth Root of a positive real number:-
If 'a' is a positive real number and n is a positive integer then the principal nth root of a is the unique positive real number x such that x power n = a.
The principal nth root of a positive real number a is denoted by a power 1/n or, n√a.
For example-
(8/27) power 1/3 = 2/3 (as [2/3]³ = 2/3 × 2/3 × 2/3)
Principal nth root of a negative real number:-
If a is a negative real number and n is an odd positive integer, then the principal nth root of a is defined as -|a| power 1/n i.e the principal nth root of a is minus of the principal nth root of |a|.
For example-
(-8) power 1/3 = -8 power 1/3 = -2
Laws of Rational Exponents
Rationalisation
After knowing all the concepts about Exponents, now we are going about Rationalisation.
Rationalisation is nothing but simplification
of expressions containing square roots in denominators by rationalising the denominators.
So before going into the concept, we have to remember some identities so that we can easily solve questions.
Sometimes we come across expressions containing square roots in their denominators. Addition, Subtraction, multiplication and division of such expressions is convenient if their denominators are free from square roots. To make the denominators free from square roots, we multiply the numerator and denominator by an irrational number.
Such a number is called Rationalisation factor.
For example-
It follows from the above example that the rationalisation factor for 1/√a is √a and for 1/a±b the rationalisation factor is a -+√b
Rationalisation factor for 1/√a±√b is √a -+ √b.
If you have still any doubts then feel free to comment us.
So first come to Exponents of real numbers-
It is nothing but the study of powers of rational numbers and the laws obeyed by itself.
Integral Exponents of a Real Number:-
POSITIVE INTEGRAL POWER-
For any real number 'a' and a positive integer n, we define a power n as
a power n = a×a×a×......×a (n factors)
a power n is called the nth power of a. The real number a is called the base and n is called the exponent (index) of the nth power of a.
For example-
2³ = 2×2×2 = 8
NEGATIVE INTEGRAL POWER:-
For any non-zero real number 'a' and a positive integer n, we define
a power -n = 1/a power n
For example-
5-³ = 1/5³ = 1/5×5×5 = 1/125
Laws of Integral Exponents
First Law:- If a is any real number and m, n are positive integers, then a power m × a power n = a power (m+n).
For example-
5⁴ × 5³ = 5⁴+³ = 5 power 7
Second Law:- If a is a non-zero real number and m, n are positive integers, then a power m/a power n = a power (m-n).
For example-
3 power 5 ÷ 3² = 3 power (5-2) = 3³.
Third Law:- If a is any real number and m, n are positive integers, then (apower m) power n = a power m×n = (a power n) power m.
For example-
(3²) power 5 = 3 power 2×5 = 3 power 10
Fourth Law:- If a, b are real numbers and m, n are positive integers, then
1. (ab) power n = a power n × b power n
2. (a/b) power n = a power n/b power n, where b is not equal to 0.
For example-
1. 6⁴ = (2×3)⁴ = 2⁴ × 3⁴
Rational Exponents of a real number
Principal nth Root of a positive real number:-
If 'a' is a positive real number and n is a positive integer then the principal nth root of a is the unique positive real number x such that x power n = a.
The principal nth root of a positive real number a is denoted by a power 1/n or, n√a.
For example-
(8/27) power 1/3 = 2/3 (as [2/3]³ = 2/3 × 2/3 × 2/3)
Principal nth root of a negative real number:-
If a is a negative real number and n is an odd positive integer, then the principal nth root of a is defined as -|a| power 1/n i.e the principal nth root of a is minus of the principal nth root of |a|.
For example-
(-8) power 1/3 = -8 power 1/3 = -2
Laws of Rational Exponents
Rationalisation
After knowing all the concepts about Exponents, now we are going about Rationalisation.
Rationalisation is nothing but simplification
of expressions containing square roots in denominators by rationalising the denominators.
So before going into the concept, we have to remember some identities so that we can easily solve questions.
Sometimes we come across expressions containing square roots in their denominators. Addition, Subtraction, multiplication and division of such expressions is convenient if their denominators are free from square roots. To make the denominators free from square roots, we multiply the numerator and denominator by an irrational number.
Such a number is called Rationalisation factor.
For example-
It follows from the above example that the rationalisation factor for 1/√a is √a and for 1/a±b the rationalisation factor is a -+√b
Rationalisation factor for 1/√a±√b is √a -+ √b.
If you have still any doubts then feel free to comment us.