Rational Exponent

Rational numbers are numbers of the form `p/q ", where " q !=0`

Rational exponent expressions will have the fractional numbers as power. The general  form is `x^(b/c),` here `(b/c)`   is the rational exponent . We can perform all the types of arithmetic operations with these expressions . It may have positive or negative rational exponents.

When there is a rational exponent `b/c` the numerator part `b` exponential part and denominator `c` is the index or root number.

The number `x^(b/c) = root(c)(x^b)`

Steps involved in solving expressions with rational exponents:

First find the prime factors of the base

Second express the prime factors in exponent form

Apply the rules of multiplication of two exponents to the same base

Simplify the exponents

Simplify the exponent form to ordinary form


Types of Rational Exponents

Below are the types of rational exponents:

Positive rational exponents:

If x is a positive rational number, and also  m =` p/q` is a positive rational exponent, then we define xp/q as the qth root of xp

i.e.,      x p/q =(xp) 1/q

For ex:         8 5/3 = (85) 1/3 = (32768) 1/3 = 32

Negative exponents:

If m is a positive integer in addition to x is a non-zero rational number, then

x-m = (1/xm) = (1/x) m

i.e., x -m be the reciprocal of xm or the mth power rule of the reciprocal of x.

For ex:         ` 8^(-2/3)`

Sol:           8 -2/3 = `(1/8)^( 2/3)`

= [ ( 1/8) 1/3 ] 2

= (1/2)2 , since (1/2) 3 = `1/8`

=` 1/4.`

Laws of rational exponents

Law (1) : If x > 0 is a rational number and m and n are rational exponents, then

xm × xn = xm + n

Law (2) : For a rational number x > 0 and rational exponents m and n,

xm ÷ xn = x m – n

Law (3) : If x is a rational number, x > 0 and m and n are rational exponents, then

(xm)n = x mn

Law (4) : If x and y are rational numbers, x,y > 0 and m is a rational exponent, then

xm × y m = (x × y)m

These are the laws which we can use to solve the rational exponent expreesions

Solved Examples

Ex 1: Solve  `(125x^(3/4) y3)^(1/3)`

Sol: Step 1: We first take the exponent to each term

`125^(1/3) x^((3/4)xx(1/3)) y^(3xx1/3)`

Step 2: We find the prime factors of 125 and write it in exponent form

125 = 5 x5 x 5 = `5^3`

Step 3: We simplify the exponents of x and y

`3/4 xx1/3 = 1/4`

`3 xx1/3 =1`

Step4: `(5^3)^(1/3) x^(1/4) y`

Step 5: Simplify the exponent of 5

`5x^(1/4)y`