Exponent Rules

The explanations and examples below on exponent rules follow on from the Power (Exponents and Bases) page which you might want to start with.

The examples on simplifying exponential expressions show when and how the above exponent rules are used.

Exponents: The Product Rule

The product rule for exponents states that when we multiply exponential expressions having the same base, we can add the exponents and keep the base unchanged.

Product RuleIf m and n are natural numbers, and a is a real number, then
am x an = am + n
ExampleRewrite 4243 using a single base and exponent.
The product rule states that am x an = am + n
Applying the rule to the expression and simplifying we get
42 x 43 = 42 + 3 = 45
42 = 4 x 4 and 43 = 4 x 4 x 4
Think about it this way so42 x 43 = (4 x 4) x (4 x 4 x 4) = 45

Zero as an exponent

According to the zero exponent rule, any nonzero number raised to the power 0 equals 1.

Zero RuleIf a is any nonzero number, then
a0 = 1
ExampleEvaluate the expression (-2)0
The zero exponent rule states that a0 = 1
Applying the rule to the expression we get
(-2)0 = 1

Quotient rule for exponents

We make use of quotient rule when dividing exponents with the same base. The quotient rule for exponents states that if we divide exponents with the same base, then we can subtract the exponents and keep the base unchanged.

Quotient RuleIf b is any nonzero real number, and m and n are nonzero integers, then
bm= bm – nbn
ExampleSimplify the expression

The exponents have the same base so we can directly apply the quotient rule and rewrite the given expression as shown below.


 42 – 21 = 16 – 2 = 14

Negative Exponent Rule

We use the negative exponent rule to change an expression with a negative exponent to an equivalent expression with a positive exponent. The rule states that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power. In other words, an expression raised to a negative exponent is equal to 1 divided by the expression with the sign of the exponent changed.

Negative Exponent RuleFor any nonzero real number b, and any whole number m,
b-m =1bm 
ExampleWrite the expression 6a-2 without a negative exponent.
Apply the rule to rewrite the term a-2 since a is raised to a negative power. Write the reciprocal of a and raise it to the opposite power of –2, which is 2.

So, 6a-2 written without negative exponent is 6/a2 

Simplifying Exponential Expressions – Putting it altogether

We can simplify exponential expressions using a suitable combination of the rules and properties above.

Question: Simplify
Step 1
We know that the power rule for exponents states that if x is a real number, and m and n are integers,
then (xm)n = xmn
1. Apply the power rule to both the numerator and the denominator, and then simplify.

Notation: a dot is used above to represent multiplication
Step 2
The quotient rule for exponents states that if m and n are natural numbers, and a is a real number, then
2. Rewrite the expression using the quotient rule for exponents.
Step 3
The negative exponent rule states that for any nonzero real number, a, and any whole number m, then
3. Use the negative exponent rule to rewrite the terms without negative exponents and simplify.