Objective: Simplify expressions using the properties of exponents.
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!)
\begin{align*}
a^3a^ 2 \amp \quad \text{ Expand exponents to multiplication problem } \\
(aaa)(aa) \amp\quad \text{ Now we have 5a ' s being multiplied together } \\
a^5 \amp\quad \text{ Our Solution }
\end{align*}
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
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.
\begin{align*}
2x^ 3 y ^5 z \cdot 5xy^2 z ^3 \amp \quad \text{ Multiply $2 \cdot 5$, add exponents on $x$,$ y$ and } z \\
10x ^4 y ^7 z ^4 \amp\quad \text{ Our Solution }
\end{align*}
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.
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.
\begin{align*}
( a^ 2)^ 3 \amp \quad \text{ This means we have a $2$ three times } \\
a ^2 \cdot a ^2 \cdot a ^2 \amp\quad \text{ Add exponent } \\
a^6 \amp\quad \text{ Our solution }
\end{align*}
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.
\begin{align*}
(ab)^ 3 \amp \quad \text{ This means we have $(ab)$ three time } \\
(ab)(ab)(ab) \amp\quad \text{ Three $a$'s and three $b$'s can be written with exponents } \\
a^ 3 b ^3 \amp\quad \text{ Our Solution }
\end{align*}
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
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.
\begin{align*}
\left( \frac{a}{ b} \right)^3 \amp \quad \text{ This means we have the fraction three timse } \\
\left(\frac{ a}{ b} \right) \left(\frac{ a}{ b}\right) \left(\frac{ a}{ b}\right) \amp\quad \text{ Multiply fractions across the top and bottom, using exponents } \\
\frac{a ^3 }{b ^3} \amp\quad \text{ Our Solution }
\end{align*}
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.
\begin{align*}
(x ^3 yz^2 )^ 4 \amp \quad \text{ Put the exponent of $4$ on each factor, multiplying power } \\
x^{12}y ^4 z^8 \amp\quad \text{Our solution }
\end{align*}
\begin{align*}
\left(\frac { a ^3 b }{c^ 8d ^5}\right)^2 \amp \quad \text{ Put the exponent of $2$ on each factor, multiplying powers } \\
\frac {a^ 6 b ^2} {c ^{16}d^{10} } \amp\quad \text{ Our Solution }
\end{align*}
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.
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.
