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 nonzero 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 nonzero 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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