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
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)
= (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)
= 3
4. Find 1251/3
Solution:
1251/3
= ∛125
= [(5) 3]1/3
= (5)
= 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)
= 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)
= (5/3)4
= (5 × 5 × 5 × 5)/(3 × 3 × 3 × 3)
= 625/81
● Exponents
Integral Exponents of a Rational Numbers
● Exponents - Worksheets
8th Grade Math Practice
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