Simplifying Exponents

To understand how to simplify exponents, we first need to revise the concept of exponents. An exponent tells us the number of times a number is multiplied by itself in the given expression. 
For example:

3 × 3 × 3 × 3 × 3 = 3⁵

Expressing a number using exponents is easier and clearer than expanding it.
In the general algebraic form, it can be represented as aᵐ, where a is multiplied by itself m times.
Simplifying exponents is nothing but writing expressions with exponents in a simpler form.
 

As discussed above, exponents show how many times the base number is multiplied by itself. An exponent has two components: the base and the power (also called the exponent).

The exponent ‘n’ of the base ‘a’ is written as: aⁿ
Here, the base ‘a’ denotes the number that is being multiplied.
The exponent ‘n’  indicates the number of times the base is multiplied by itself.

Let’s look at the key rules that help us simplify expressions with exponents. A few of these rules are shown in the table below:
 

To simplify exponents, we follow specific rules known as the laws of exponents. These are: 

• Product Rule → aᵐ × aⁿ = aᵐ⁺ⁿ
• Quotient Rule → aᵐ / aⁿ = aᵐ⁻ⁿ
• Zero Exponent Rule → a⁰ = 1
• Identity Rule → a¹ = a
• Negative Exponent Rule → a⁻ᵐ = 1 / aᵐ
• Power of a Power Rule → (aᵐ)ⁿ = aᵐⁿ
• Power of a Product Rule → (ab)ᵐ = aᵐ × bᵐ
• Power of a Quotient Rule → (a/b)ᵐ = aᵐ / bᵐ
   

There are mainly two cases when simplifying exponents with different bases:

• When numbers have different bases but identical exponents.

• When both the bases and the exponents are different.

How to Simplify Exponents With Different Bases and Same Power?

To simplify exponents with different bases and the same power, divide or multiply the bases first, then apply the common exponent to the result.
52/ 102 = (5/10)2 = (½)2 = 1/4.

How to Simplify Exponents With Different Bases and Different Powers? 

For exponential terms with different bases and powers, first simplify each term separately, then perform multiplication or any other operation.
Example:
3² × 2³
= 9 × 8
= 72

Simplifying Exponents in Fractions

Exponents written in fractional form are known as fractional exponents or rational exponents. They represent roots and powers in a more generalized way.
To simplify expressions with exponents in fractions, we apply the quotient rule by simplifying the numerator and denominator separately.
(10x4y3) ÷ (2x2y2)
= (10 ÷ 2) (x4 ÷ x2) (y3 ÷ y2)
= 5x4 –2 y3 – 2
= 5x2y

Simplifying Negative Exponents

A negative exponent means to take the reciprocal and then apply the corresponding positive exponent. It can be written as:
a–n = 1/ an
After converting the exponent to a positive value, we can simplify any remaining factors if needed.
7q–2 = 7/ q2

Simplifying Rational Exponents

Rational exponents are exponents written as fractions. They represent both powers and roots. To simplify rational exponents, we can either apply the exponent rules or convert them to their equivalent root form.
For example:
b4/2 ÷ b1/2
= b(4/2 –1/ 2) = b3/2 
⇒ √b3
 

Simplifying exponents is a significant concept in math that helps students understand and solve complex expressions easily. This concept is used in various other fields beyond math. Let’s take a look at their widespread applications in the real world.

• Exponents are used in space science to express large numbers like the distances between planets or stars.

• This concept is widely used in the banking sector to calculate the interest on money deposited in savings accounts.

• Other fields like chemistry or physics use exponents in formulas such as Ohm’s Law, or to calculate energy using powers.