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 = 5^{3 × 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 9^1/2

Solution:

9^1/2

= √(2&9)

= [(3)²]^1/2

= (3)^{2 × 1/2}

= 3



4. Find 125^1/3

Solution:

125^1/3

= ∛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/x^m = (1/x)^m, i.e., x^-m is the reciprocal of x^m.

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

= 1/9^1/2

= (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

● Exponents

Exponents

Laws of Exponents

Rational Exponent

Integral Exponents of a Rational Numbers

Solved Examples on Exponents

Practice Test on Exponents

● Exponents - Worksheets

Worksheet on Exponents

8th Grade Math Practice

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