# 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

**x**

^{n}= y ; then y^{1/n}is also written as n√yand 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

• Laws of Exponents

• Rational Exponents

• Integral Exponents

• Scientific notation

• Solved examples on Scientific Notation

• Solved Examples on Exponents