Exponent and Power

In the expression 10⁹, 9 is called an exponent. It tells us that 10 should be multiplied by itself 9 times. Let’s consider another example, 3². Here, the exponent is 2, and it tells us that 3 should be multiplied by itself twice. So, 3² = 3  3 = 9.
 

In the expression 5³, 5 is the base, 3 is the exponent, and the whole expression (5³) is called power. Although there is a common misconception that power is the same as exponent, we should always remember that power and exponent are two different things.

• In plain terms, an is called a to the nth power or the nth power of a.
  
   
• Here:

a is the base.

n is the exponent or index (indicating how many times the base is multiplied).

The entire expression an  is the power.

So, in “3 to the 4th power,” the power points to the full expression 3⁴ (which equals 3×3×3×3 = 81), not just the “4.”

Sometimes, students might get confused between exponent and power. Some may even think that they are one and the same. However, they are two different mathematical terms with different functions. Let’s look at their differences in the table below:
 

1. Multiplication Law

If two powers with the same base are multiplied, then the exponents are added.
Example:

a^m x aⁿ = a^m+n
2³  x 2² = 2³⁺² = 25

2. Division Law

If two powers with the same base are divided, then the exponents are subtracted.

Example:

a^m x aⁿ=a^m-n

2⁵/2²=2^5-2=23

3. Negative Exponent Law

A negative exponent indicates the reciprocal of the base raised to the positive exponent.

Example:

a^-n=1/aⁿ

2^-3=1/2³=1/8
 

Exponent and power have many real-life applications in various fields. Let’s take a look at some of those applications. 

 

• Biology: In biology, bacterial population growth follows N = N0⋅2t, where N0 is the initial population and t is time, modeling exponential growth.
  
   
• Art & Design: Artists use scaling exponents to create self-similar branching patterns in paintings and designs.
  
   
• Architecture: Architects use exponents and powers to calculate area and volume. They also use them to evaluate material strength, scale models accurately, and design lighting and acoustics.
  
   
• Nature: In nature, fractal tree branching follows power laws, where branch diameters scale exponentially to maintain structural efficiency.
  
   
• Urban & Landscape Design: Biophilic design buildings use exponential scaling and power-law patterns inspired by nature to evoke wellbeing and reduce stress.