# Rational Exponent

In rational exponent there are positive rational exponent and negative rational exponent.

### Positive Rational Exponent:

We know that 2³ = 8. It can also be expressed as 8$$^{\frac{1}{3}}$$ = 2.

In general if x and yare non-zero rational numbers and m is a positive integer such that xᵐ = y then we can also express it as y$$^{\frac{1}{m}}$$ = x but we can write y$$^{\frac{1}{m}}$$ = $$\sqrt[m]{y}$$ and is called mᵗʰ root of y.

For example, y$$^{\frac{1}{2}}$$ = $$\sqrt[2]{y}$$, y$$^{\frac{1}{3}}$$ = ∛y, y$$^{\frac{1}{4}}$$ = ∜y, etc. If x is a positive rational number then for a positive ration exponent p/q we have x₀can be defined in two equivalent form.

x$$^{\frac{p}{q}}$$ = $$(x^{p})^{\frac{1}{q}}$$ = $$\sqrt[q]{x^{p}}$$ is read as qᵗʰ root of xᵖ

x$$^{\frac{p}{q}}$$ = $$(x^{\frac{1}{q}})^{p}$$ = $$(\sqrt[q]{x})^{p}$$ is read as pᵗʰ power of qᵗʰ root of x

For example:

1. Find (125)$$^{\frac{2}{3}}$$

Solution:

(125)$$^{\frac{2}{3}}$$

125 can be expressed as 5 × 5 × 5 = 5³

So, we have (125)2/3 = (53)2/3 = 53 × 2/3 = 52 = 25

2. Find (8/27)4/3

Solution:

(8/27)4/3

8 = 23 and 27 = 33

So, we have (8/27)4/3 = (23/33)4/3

= [(2/3) 3]4/3

= (2/3)3 × 4/3

= (2/3) 4

= 2/3 × 2/3 × 2/3 × 2/3

= 16/81

3. Find 91/2

Solution:

91/2

= √(2&9)

= [(3)2]1/2

= (3)2 × 1/2

= 3

4. Find 1251/3

Solution:

1251/3

= ∛125

= [(5) 3]1/3

= (5) 3 × 1/3

= 5

### Negative Rational Exponent:

We already learnt that if x is a non-zero rational number and m is any positive integer then x-m = 1/xm = (1/x)m, i.e., x-m is the reciprocal of xm.

Same rule exists of rational exponents.

If p/q is a positive rational number and x > 0 is a rational number

Then x-p/q = 1/xp/q = (1/x) p/q, i.e., x-p/q is the reciprocal of xp/q

If x = a/b, then (a/b)-p/q = (b/a)p/q

For example:

1. Find 9-1/2

Solution:

9-1/2

= 1/91/2

= (1/9)1/2

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

= (1/3)2 × 1/2

= 1/3

2. Find (27/125)-4/3

Solution:

(27/125)-4/3

= (125/27)4/3

= (53/33)4/3

= [(5/3) 3]4/3

= (5/3)3 × 4/3

= (5/3)4

= (5 × 5 × 5 × 5)/(3 × 3 × 3 × 3)

= 625/81

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Exponents

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Integral Exponents of a Rational Numbers

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Practice Test on Exponents

Exponents - Worksheets

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