Exponent and Power

In the expression xⁿ, n is called an exponent. It tells us that 'x' should be multiplied by itself 'n' times. Here, x is called the base.
 

Let’s consider an example, 3². Here, the exponent is 2, and it tells us that 3 should be multiplied by itself twice.

So, 3² = 3 × 3 = 9.

Parent Tip: Give your child 2 candies four times. Now, ask how many candies does he/she has.
It will be 2 × 2 × 2 × 2. Since, 2 is multiplied 4 four times, it can be written as 2⁴.

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:
 

Here are some basic laws of exponents or exponents rules:

1. Multiplication Law
   If two powers with the same base are multiplied, then the exponents are added.
   Expression: \(a^m \times a^n = a^{m+n}\)
   Example: \(2^3 \times 2^2 = 2^{3+2} = 25\)
    
2. Division Law
   If two powers with the same base are divided, then the exponents are subtracted.
   Expression: \(a^m \div a^n=a^{m-n}\)
   Example: \(2^5 \div 2^2=2^{5-2} = 2^3\)

    

3. Negative Exponent Law
   A negative exponent indicates the reciprocal of the base raised to the positive exponent.
   Expression: \(a^{-n}= \frac{1}{a^n}\)
   Example: \(2^{-3}= \frac{1}{2^3}\)
    

4. Power of Power
   If you raise an exponent to another exponent, multiply them.
   Expression: \(({a^m})^n = a^{m \times n}\)
   Example: \(({2^2})^3 = 2^{2 \times 3} = 2 ^{6}\)

Here is the summarized table for basic exponents rule:

Let's look at some essential tips and tricks to help you grasp the concepts of exponents and power.
 

1. Learn that any number to the power of 0 is equal to 1.
    
2. Remember, exponents are like repeated multiplication. It tells you how many times a number is multiplied.
    
3. Any number to the power of 1 is equal to itself.
    
4. If a number has even exponents, it will always give a positive result.
    
5. A negative number to the power of another negative number will give you a negative result.

Parent Tip: You can use an exponent calculator to verify your child's calculation. Help memorize exponents laws to your child. You can also use real life items for better visualization.

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

 

1. Biology: In biology, bacterial population growth follows N = N0⋅2t, where N0 is the initial population and t is time, modeling exponential growth.
   
    
2. Art & Design: Artists use scaling exponents to create self-similar branching patterns in paintings and designs.
   
    
3. 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.
   
    
4. Nature: In nature, fractal tree branching follows power laws, where branch diameters scale exponentially to maintain structural efficiency.
   
    
5. Urban & Landscape Design: Biophilic design buildings use exponential scaling and power-law patterns inspired by nature to evoke wellbeing and reduce stress.