Class 8 Maths Exponents and Powers Powers with Negative Exponents

Powers with Negative Exponents

For any non-zero integer x, x–n = 1/xn, where n is a positive integer and x–n is the multiplicative inverse

of xn.

Problem: Evaluate:

(i) 3-2                                                          (ii) (-4)-2                                                   (iii) (1/2)-5

Solution:

(i) 3-2 = 1/32 = 1/9                                  [a-m = 1/ am]

(ii) (-4)-2 = 1/42 = 1/16                          [a-m = 1/ am]

(iii) (1/2)-5 = (2/1)5 = 25 = 32                [a-m = 1/ am]

Laws of Exponents

Numbers with negative exponents obey the following laws of exponents.

(a) am * an = am+n                                   (b) am / an = am-n                                         (c) (am)n = amn

(d) am * bm = (ab)m                                (e) a0 = 1                                                      (f) am / bm = (a/b)m

(g) (a/b)-m = (b/a)m

Here, a and b are any non-zero integers and m and n are natural numbers.

Problem: Simplify and express the result in power notation with positive exponent:

(i) (-4)5 ÷ (-4)8               (ii) (1/23)2            (iii) (-3)4 * (5/3)4           (iv)  (3-7 * 3-10) * 35             (v) 2-3 * (-7)3

Solution:

(i) (-4)5 ÷ (-4)8 = (-4)5-8                                             [am ÷ an = am-n]

= (-4)-3

= 1/(-4)                                         [a-m = 1/ am]

= -1/64

(ii) (1/23)2 = 12/(23)2                                               [(a/b)m = am/bm]

= 1/ 23*2                                                 [(am)n = am*n]

= 1/26

= 1/64

(iii) (-3)4 * (5/3)4 = (-3)4 * (54/34)                        [(a/b)m = am/bm]

= (3)4 * (54/34)                         [(-a)m = am when m is an even number]

= (3)4-4 * 54

= 54

(iv)  (3-7 * 3-10) * 35 = 3-7-10+5                                [am * an = am+n]

= 3-17+5

= 3-12

= 1/312                                 [a-m = 1/ am]

(v) 2-3 * (-7)-3 = 1/23 * 1/(-7)-3                            [a-m = 1/ am]

= 1/{(-7)3 * 23 }

= 1/(-7 * 2)3                                  [am * bm = (a * b)m]

= 1/(-14)3

= -1/(14)3                                       [(-a)m = -am when m is an odd number]

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