Indices and Logarithms

1. Laws of Indices

Laws of Indices (Exponents)

Indices (or exponents) are used to express repeated multiplication of a number. If $a$ is a real number and $m, n$ are integers, then the laws of indices are as follows:

1. Product Law

a^m \times a^n = a^{m+n}

Example:

2^3 \times 2^4 = 2^{3+4} = 2^7

2. Quotient Law

\frac{a^m}{a^n} = a^{m-n}, \quad \text{where } a \neq 0

Example:

\frac{5^6}{5^2} = 5^{6-2} = 5^4

3. Power of a Power Law

(a^m)^n = a^{m \times n}

Example:

(3^2)^4 = 3^{2 \times 4} = 3^8

4. Power of a Product Law

(ab)^m = a^m \times b^m

Example:

(2 \times 3)^4 = 2^4 \times 3^4

5. Power of a Quotient Law

\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}, \quad \text{where } b \neq 0

Example:

\left( \frac{4}{5} \right)^3 = \frac{4^3}{5^3} = \frac{64}{125}

6. Zero Index Law

a^0 = 1, \quad \text{where } a \neq 0

Example:

7^0 = 1

7. Negative Index Law

a^{-m} = \frac{1}{a^m}, \quad \text{where } a \neq 0

Example:

3^{-2} = \frac{1}{3^2} = \frac{1}{9}

8. Fractional Indices (Roots Law)

a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m

Example:

16^{\frac{1}{2}} = \sqrt{16} = 4

27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9

Examples Using Multiple Laws

Simplify $2^3 \times 2^{-1}$
- Using the product law:
  $2^{3+(-1)} = 2^2 = 4$
Simplify $\frac{5^4}{5^6}$
- Using the quotient law:
  $5^{4-6} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
Simplify $(4^2)^3$
- Using the power of a power law:
  $4^{2 \times 3} = 4^6 = 4096$