# Laws of Indices

We will discuss here about the different Laws of Indices.

If a, b are real numbers (>0, ≠ 1) and m, n are real numbers, following properties hold true.

(i) am × an = am + n

(ii) a-m = $$\frac{1}{a^{m}}$$

(iii) $$\frac{a^{m}}{a^{n}}$$ = am – n = $$\frac{1}{a^{m - n}}$$

(iv) (am)n = amn

(v) (ab)n = an ∙ bn

(vi) a0 = 1

(vii) if am = an then m = n.

(viii) if an = bn, a ≠ b then n = 0.

Note: Some of the above properties hold true for any two real numbers a, b. Laws (i) to (v) hold true for any two real numbers a, b. Also note that 10 = 1.

Problems on knowledge and use of the properties of indices:

1. Determine the numerical value for each of the following (not containing exponents):

(i) 64

(ii) (-5)-4

(iii) 90

(iv) ($$\frac{1}{4}$$)-5 (vi) ($$\frac{1}{3}$$)0

Solution:

(i) 64 = 6 × 6 × 6 × 6 = 1296; [Using the definition of power/exponent].

(ii) (-5)-4 = $$\frac{1}{(-5)^{4}}$$; [Using the property of indices].

= $$\frac{1}{(-5) × (-5) × (-5) × (-5)}$$ ; [Using the definition of power].

= $$\frac{1}{25 × 25}$$

= $$\frac{1}{625}$$

(iii) 90 = 1; [Using the property of indices: here 9 ≠ 0].

(iv) ($$\frac{1}{4}$$)-5 = (4-1)-5 = 4(-1) × (-5) = 45 = 1024

(vi) ($$\frac{1}{3}$$)0 = 1; [Using the property of indices: here $$\frac{1}{3}$$ ≠ 0].