8 Basic Exponent Rules

In PEMDAS, exponents are the second letter in the acronym. They are commonly found in many equations, and here are 8 basic rules to consider when operating with exponents.

1. Zero Power Property

The first exponent rule is quite simple and involves expressions in which the exponent is zero on a non-zero base. In these instances where any non-zero quantity is raised to the power of zero, the value of the expression is equal to 1.

This may seem odd at first but makes sense mathematically. If we raise 3 to the power of 3, it is the same as 1 * 3*3*3. 3 to the power of 2 becomes 1 * 3*3, and 3 to the power of 1 is 1 * 3. Continuing with the pattern, 3 to the power of 0 becomes just 1 because 0 times of 3 is just 0, so nothing needs to be multiplied to 1.

2. Negative Exponent

Although typically, exponents are positive numbers, occasionally there will be an exponent which is a negative quantity. In this case, if the non-zero base is being raised to any negative exponent, it is equal to one divided by the base raised to a positive exponent. This positive exponent is the original negative exponent negated, in turn resulting in a positive value.

3. Product of Powers (ADD)

Have you seen problems where two or more powers with the same base are being multiplied together? The product of powers rule states that when this occurs, the expression is equal to the same, common base raised to the sum of the exponents for all of the powers being multiplied.

If you had 42 * 43, this would be the same as 4*4 * 4*4*4 by the definition of an exponent, which we can see is 45 as there are five 4’s multiplying together, which is the same as 42+3.

4. Quotient of Powers (SUBTRACT)

Quotient of powers states that when there is the quotient of two exponential terms, both with the same base and exponents, the expression is equal to the base raised to the difference of the exponents. In other words, keep the base that is in common but subtract the exponents.

The concept behind this is very similar to that of the product of powers rule because they are so similar – if we have 33 / 32, this is the same as 3*3*3/3*3. Canceling out to simplify, this becomes just 3 (3/3 is 1 so we can take it out), which is the same as 31 or 33-2.

5. Power to a Power (MULTIPLY)

When a power is raised to another power, this rule helps us do operations correctly by stating how the expression is equal to the base raised to the product of the powers, by multiplying these powers.

6. Product of Power/Power of Product (DISTRIBUTION)

Commonly associated with distribution, there is a connection between the product of power rule and the distributive property! This rule works when an expression is set up so that there are multiple terms being multiplied and all are being raised to the same power – we can simplify this so that we can just write the product of each factor raised to that same power.

7. Quotient to a Positive Power

When a quotient is raised to a positive power, it is equal to the quotient of each base raised to that common power. This is very similar to rule #6, except with division (there is a quotient) instead of multiplication.

8. Negative Power of a Quotient

Finally, rule #8, or negative power of a quotient states that when a quotient is raised to a negative power, it is equal to the reciprocal of the quotient raised to the opposite, positive power. This is really a combination of the other exponent rules, as it blends together both the quotient to a positive power rule and the negative exponent rule.