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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 Rule If m and n are natural numbers, and a is a real number, then
am x an = am + n
Example Rewrite 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 so 42 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 Rule If a is any nonzero number, then
a0 = 1
Example Evaluate 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.

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.

Simplifying Exponential Expressions – Putting it altogether

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