We shall be dealing with the positive and negative integral exponents of a rational numbers.
Let a/b be any rational number and n be a positive integer. Then,
(a/b)ⁿ = a/b × a/b × a/b × ……. n times
= (a × a × a ×…….. n times )/( b × b × b ×……….. n times )
= aⁿ/bⁿ
Thus (a/b)ⁿ = aⁿ/bⁿ for every positive integer n .
For example:
Evaluate:
(i) (3/5)³
= 3³/5³
= 3 × 3 × 3/5 × 5 × 5
= 27/125
(ii) (-3/4)⁴
= (-3)⁴/4⁴
= 34/44
= 3 × 3 × 3 × 3/4 × 4 × 4 × 4
= 81/256
(iii) (-2/3)⁵
= (-2)⁵/3⁵
= (-2)⁵/3⁵
= -2 × -2 × -2 × -2 × -2/3 × 3 × 3 × 3 × 3
= -32/243
Let a/b be any rational number and n be a positive integer.
Then, we define, (a/b)\(^{-n}\) = (b/a)ⁿ
For example:
(i) (3/4)\(^{-5}\)
= (4/3)⁵
(ii) 4\(^{-6}\)
= (4/1)\(^{-6}\)
= (1/4)⁶
Also, we define, (a/b) = 1
Evaluate:
(i) (2/3)\(^{-3}\)
= (3/2)³
= 3³/2³
= 27/8
(ii) 4\(^{-2}\)
= (4/1)\(^{-2}\)
= (1/4)²
= 1²/4²
= 1/16
(iii) (1/6)\(^{-2}\)
= (6/1)²
= 6²
= 36
(iv) (2/3) = 1
The positive and negative integral exponents of a rational numbers are explained here with examples.
● Exponents
Integral Exponents of a Rational Numbers
● Exponents - Worksheets
8th Grade Math Practice
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