Rational Exponents

Rational Exponents means the exponent in p/q form. The exponent may be positive or negative. Positive rational-exponent
3
2 = 9 ⇒ 9 1/2 = 3.

4
2 = 16 ⇒ 16 1/2 = 4

5
3 = 125 ⇒ 125 1/3 = 5

In general, if n is any positive integer greater than one; x and y be rational numbers such that
xn = y ; then y1/n is also written as n√y

and read as “nth root of y”. Here, n√ is called a radical, n is called the index of the radical and y is called the radicand.
The number written as y
1/n is called its exponential form.

Express the following in the exponential form :

1) √7 = 7
1/2

2) 7√250 = (250)
1/7

3) 4√(2/3) = (2/3)
1/4

Write the following as radicals

1) 3
1/2
= √3

2) (101)
1/3
= 3√(101)

3) (3/4)
1/5
= 5√(3/4)

Rational number as Exponents

Let ‘a’ be any positive rational number and (m/n) be a positive rational number in the lowest form; then we write


Examples :

1) (9)
3/2
= ( 9
3 ) 1/2
= √729
= 27

2) (8)
2/3
= (8
2 ) 1/3
=(64)
1/3
= 4

Evaluate :

1) (125)
2/3
= ( 5 x 5 x 5)
2/3
= [ ( 5
3 )] 2/3
= 5
3 x 2/3
= 5
6/3
= 5
2
= 5 x 5
= 25

2) (81)
1/4
= (3 x 3 x 3 x 3 )
1/4
= [(3
4 ] 1/4
= 3
4 x1/4
= 3
4/4
= 3
1
= 1

Negative rational exponents


Examples :

1) (4)
-3/2
= 1/(4
3/2 )
= 1/(4
3 ) 1/2
= 1/√64
= 1/8

2) (27)
-2/3
= 1/(27)
2/3
= 1/(27
2 ) 1/3
= 1/3√729
= 1/9

3) (49)
-3/2
= 1/(49)
3/2
= 1/(49
3 ) 1/2
= 1/√(49 x 49 x 49)
= 1/√(49 x 49 x 7 x 7)
= 1/(49 x 7)
= 1/343

Exponents

Laws of Exponents
Rational Exponents
Integral Exponents
Scientific notation
Solved examples on Scientific Notation
Solved Examples on Exponents

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