College Preparatory Mathematics

Section 5.1 Exponent Properties

Problems with exponents can often be simplified using a few basic exponent properties. Exponents represent repeated multiplication. We will use this fact to discover the important properties.

World View Note: The word exponent comes from the Latin “expo” meaning out of and “ponere” meaning place. While there is some debate, it seems that the Babylonians living in Iraq were the first to do work with exponents (dating back to the 23rd century BC or earlier!)

A quicker method to arrive at our answer would have been to just add the exponents: \(a ^3a ^2 = a^{3+2 }= a^ 5 \) This is known as the product rule of exponents

Product Rule of Exponents: a^ma^ n = a^{m+n}

The product rule of exponents can be used to simplify many problems. We will add the exponent on like variables. This is shown in the following examples.

Rather than multiplying, we will now try to divide with exponents

A quicker method to arrive at the solution would have been to just subtract the exponents, \(a^ 5 a ^2 = a ^{5-2} = a^ 3 \text{.}\) This is known as the quotient rule of exponents.

Quotient Rule of Exponents: \(\frac{a^m}{a^n} = a^{m-n}\text{.}\)

The quotient rule of exponents can similarly be used to simplify exponent problems by subtracting exponents on like variables. This is shown in the following examples.

A third property we will look at will have an exponent problem raised to a second exponent. This is investigated in the following example.

A quicker method to arrive at the solution would have been to just multiply the exponents, \((a^2)^3 = a ^{2\cdot 3} = a^ 6\text{.}\) This is known as the power of a power rule of exponents.

Power of a Power Rule of Exponents: \((a^m)^ n= a^{mn}\)

This property is often combined with two other properties which we will investigate now.

A quicker method to arrive at the solution would have been to take the exponent of three and put it on each factor in parenthesis, \((ab)^ 3 = a^ 3 b ^3 \text{.}\) This is known as the power of a product rule or exponents

Power of a Product Rule of Exponents: \((ab)^m = a^mb^m\)

It is important to be careful to only use the power of a product rule with multiplication inside parenthesis. This property does NOT work if there is addition or subtraction.

Another property that is very similar to the power of a product rule is considered next.

A quicker method to arrive at the solution would have been to put the exponent on every factor in both the numerator and denominator, \(\left(\frac{a}{ b}\right)^ 3 = \frac{a^ 3} {b^ 3}\text{.}\) This is known as the power of a quotient rule of exponents.

Power of a Quotient Rule of Exponents: \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} \)

The power of a power, product and quotient rules are often used together to simplify expressions. This is shown in the following examples.

As we multiply exponents it’s important to remember these properties apply to exponents, not bases. An expression such as \(5^ 3\) does not mean we multiply\(5\) by \(3\text{,}\) rather we multiply\(5\) three times, \(5 × 5 × 5 = 125\text{.}\) This is shown in the next example.

In the previous example we did not put the \(3\) on the \(4\) and multiply to get \(12\text{,}\) this would have been incorrect. Never multiply a base by the exponent. These properties pertain to exponents only, not bases.

In this lesson we have discussed \(5 \) different exponent properties. These rules are summarized in the following table.

These five properties are often mixed up in the same problem. Often there is a bit of flexibility as to which property is used first. However, order of operations still applies to a problem. For this reason, it is the suggestion of the author to simplify inside any parenthesis first, then simplify any exponents (using power rules), and finally simplify any multiplication or division (using product and quotient rules). This is illustrated in the next few examples.

Clearly these problems can quickly become quite involved. Remember to follow order of operations as a guide, simplify inside parenthesis first, then power rules, then product and quotient rules.