Learn Exponent with Examples

What is Exponent?

The base n raised to the power of a is equal to the multiplication of n, a times:

n^a = n × n × … × n (a times)

There:

• “n” is the base and
• “a” is the exponent

or

An exponent refers to the number of times a number is multiplied by itself.

An example of exponent

3 to the 4th, written as 3⁴ means:

 

3⁴ is not equal to 34 = 12

Special Cases

1. Exponent is 0:

n = 1

When the exponent is zero, as in 5, the result is always equal to 1.

5 = 1

26,981 = 1

2.Exponent is negative

n^-a = 1 / n^a

When an exponent is a negative integer, the result will always be a fraction. Fractions contain a numerator and a denominator. In this case, the numerator is always 1. To calculate the denominator, assume that the negative exponent is positive, and raise the number to that power, like this:

5^-4 = 1 / 5⁴

Some Important Rules for Exponents

Product rule with the same base

n^a.n^b = n^a+b

Example:

4⁵ . 4⁶ = 4 ⁵⁺⁶ = 4 ¹¹ = 4194304

Product rule with the same exponent

n^a . m^a = (n.m)^a

Example:

26. 2⁶. 3⁶ = (23)⁶ = 6⁶ = 36

 

Quotient rule with the same base

n^a / n^b = n^a-b

Example:

4⁶ / 4⁵ = 4^6-5 = 4¹ = 4

 

Quotient rule with the same exponent

n^a / m^a = (n/m)^a

Example:

10⁶ / 5⁶ = (10/5)⁶ = 2⁶ = 64

 

Power rules #1

(n^a)^b = n^a.b

Example

(3²)³ = 3^2*3 = 3⁶ = 729

Power rule 2

Example

 = 4 ^{(2^3)} = 4⁸ = 6561

Power rule with radicals

Example