# 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} = (5

^{3})

^{2/3} = 5

^{3 × 2/3} = 5

^{2} = 25

**2. Find (8/27)**^{4/3}

Solution:
(8/27)

^{4/3}
8 = 2

^{3} and 27 = 3

^{3}
So, we have (8/27)

^{4/3} = (2

^{3}/3

^{3})

^{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}
= (5

^{3}/3

^{3})

^{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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