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) a^m × aⁿ = a^{m + n}

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

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

(iv) (a^m)ⁿ = a^mn

(v) (ab)ⁿ = aⁿ ∙ bⁿ

(vi) a⁰ = 1

(vii) if a^m = 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].

9th Grade Math

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