Law of Exponents

If a is a rational number, multiplied by itself m times, it is written as am. Here, a is called the base and m is called the exponent.

Exponential Notation

Consider the following products:

1. 7 × 7
2. 3 × 3 × 3
3. 6 × 6 × 6 × 6 × 6

In (i), 7 is multiplied twice and hence 7 × 7 is written as 72.

In (ii), 3 is multiplied three times and so 3 × 3 × 3 is written as 33.

In (iii), 6 is multiplied five times, so 6 × 6 × 6 × 6 × 6 is written as 65.

72 is read as 7 raised to the power 2 or second power of 7. Here, 7 is called base and 2 is called exponent (or index).

Similarly, 33 is read as 3 raised to the power 3 or third power of 3. Here, 3 is called the base and 3 is called exponent.

Similarly, 65 is read as 6 raised to the power 5 or fifth power of 6. Again 6 is base and 5 is the exponent (or index).

If a is a rational number, multiplied by itself m times, it is written as am. a is called the base and m is called the exponent.

Laws of Exponents

Positive Integers as Exponents

Law 1: If a is any non-zero rational number and m and n are two positive integers, then

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

Law 2: If a is any non-zero rational number and m and n are positive integers (m > n), then

$$\frac{a^m}{a^n} = a^{m-n}$$

Law 3: When n > m

$$\frac{a^m}{a^n} = \frac{1}{a^{m-n}}$$

Law 4: If a is any non-zero rational number and m and n are two positive integers, then

$$(a^{m})^{n} = a^{mn}$$

Law 5 (Zero Exponent): If a is any rational number other than zero

$$a^{0} = 1$$

Negative Integers as Exponents

$$a^{-n} = \frac{1}{a^{n}}$$