Laws of Exponents

Laws of Exponents

Laws of Exponents are short forms used to represent a number of repeated factors. We use Exponential Notation to represent repeated multiplication. When we say

2 × 2 × 2 × 2 × 2 = 2⁵--- Exponential Notation

2 is the base and 5 is the exponent.

In general for any real number a and positive integer m then a multiplied m times can be written as,

a × a × a × a × a × a ….. m times = a^m

All the Laws of Exponents can be derived from the basic laws of multiplication and division.

Multiplication of Exponents or The Product Rule for Exponents

Let us consider 3³ × 3²

3³ × 3²= 3 • 3 • 3 • 3 • 3 = 3⁵

This can be rewritten as 3³ × 3²= 3³⁺²

= 3⁵

In general, for any real number a and rational numbers, m and n we have

a^m × aⁿ= (a × a × a × a × …..m times) × (a × a × a × a…… n times)

a^m × aⁿ = m factors × n factors

a^m × aⁿ = total of (m+n) factors

a^m × aⁿ = a^m+n

The product of two exponential numbers with the same base is the base raised to the power of the sum of the exponents. Remember, the bases must be same! We cannot use this rule if the bases are different.

Power of Exponents

Consider (3⁴)²

(3⁴)²= (3⁴)× (3⁴)

= ( 3 × 3 × 3 × 3) × (3 × 3 × 3 × 3)

= 3⁸

This can be rewritten as (3⁴)²= 3^4×2= 3⁸

^{Let me show you one example on Law of Exponents:}

Example: Simplify (3⁸ × 3⁴)/3²²

Solution: Using the rule a^m × aⁿ = a^m+n

(3⁸ × 3⁴) = 3⁸⁺⁴

(3⁸ × 3⁴) = 3¹²

Now we have 3¹²/ 3²²

Using the rule a^m ÷ aⁿ = a^m-n

3¹²/ 3²²= 3^12-22 = 3^-10

The solution has a negative power since n > m.

Hence, (3⁸ × 3⁴)/3²² = 3^-10