Rational Exponents

Rational Exponents means the exponent in p/q form. The exponent may be positive or negative. Positive rational-exponent
32 = 9 ⇒ 91/2 = 3.

42 = 16 ⇒ 161/2 = 4

53 = 125 ⇒ 1251/3 = 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 y1/n is called its exponential form.

Express the following in the exponential form :

1) √7 = 71/2

2) 7√250 = (250)1/7

3) 4√(2/3) = (2/3)1/4

Write the following as radicals

1) 31/2
= √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
= ( 93)1/2
= √729
= 27

2) (8)2/3
= (82)1/3
=(64)1/3
= 4

Evaluate :

1) (125)2/3
= ( 5 x 5 x 5)2/3
= [ ( 53)]2/3
= 53 x 2/3
= 56/3
= 52
= 5 x 5
= 25

2) (81)1/4
= (3 x 3 x 3 x 3 )1/4
= [(34]1/4
= 34 x1/4
= 34/4
= 31
= 1

Negative rational exponents


Examples :

1) (4)-3/2
= 1/(43/2)
= 1/(43)1/2
= 1/√64
= 1/8

2) (27)-2/3
= 1/(27)2/3
= 1/(272)1/3
= 1/3√729
= 1/9

3) (49)-3/2
= 1/(49)3/2
= 1/(493)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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