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} × a^{n} = 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} = a^{n} ∙ b^{n}

(vi) a^{0} = 1

(vii) if a^{m} = a^{n} then m = n.

(viii) if a^{n} = b^{n}, 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 1^{0} = 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) 6^{4}

(ii) (-5)^{-4}

(iii) 9^{0}

(iv) (\(\frac{1}{4}\))^{-5}

(vi) (\(\frac{1}{3}\))^{0}

**Solution:**

(i) 6^{4} = 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) 9^{0 }= 1; [Using the property of indices: here 9 ≠ 0].

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

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

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