Indices

In the expression xn, n is called the exponent or index and x is called the base and xn is read as ‘x to the power of n’ or ‘x raised to the power n’.
eg,
1. 3⁶ = 3 × 3 × 3 × 3 × 3 × 3 = 729
4³ = 4 × 4 × 4 = 64
The expression (xm)n is read as ‘x raised to the power m whole raised to the power n’.
2. (2⁴)³ = (16)³ = 212
The expression is read as ‘x raised to the power m power n’.
3. = 264
Hence, (x^m)n¹

Laws of Indices

1. a^m × aⁿ = a^{m + n}

2. (a^m)ⁿ = a^mn

3. (ab)^m = a^m b^m

4. a⁰ = 1 if (a ¹ 0)

5. When the bases of two numbers are equal, then their powers are also equal. eg, If 3ⁿ = 3², then, n = 2.

6. When the powers of two numbers are equal, then their bases are equal. eg, If a³ = 4³, then a = 4. Where a is a positive real number and n is a rational number.