Simplifying Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify \(4·2+1\) we’d first multiply \(4·2\) to get \(8\) and then add the \(1\) to get \(9.\) A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

\(4·2+1\)

\(8+1\)

\(9\)

Suppose we have the expression \(2·2·2·2·2·2·2·2·2.\) We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write \(2·2·2\) as \({2}^{3}\) and \(2·2·2·2·2·2·2·2·2\) as \({2}^{9}.\) In expressions such as \({2}^{3},\) the \(2\) is called the base and the \(3\) is called the exponent. The exponent tells us how many factors of the base we have to multiply.

The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as “base” and the number three is labeled as “exponent”.

\(\text{means multiply three factors of 2}\)

We say \({2}^{3}\) is in exponential notation and \(2·2·2\) is in expanded notation.

Exponential Notation

For any expression \({a}^{n},a\) is a factor multiplied by itself \(n\) times if \(n\) is a positive integer.

\({a}^{n}\text{means multiply}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{factors of}\phantom{\rule{0.2em}{0ex}}a\)

At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as “base” and the letter n is labeled as “exponent”. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.

The expression \({a}^{n}\) is read \(a\) to the \({n}^{th}\) power.

For powers of \(n=2\) and \(n=3,\) we have special names.

\(\begin{array}{l}{a}^{2}\phantom{\rule{0.2em}{0ex}}\text{is read as}\phantom{\rule{0.2em}{0ex}}\text{"}a\phantom{\rule{0.2em}{0ex}}\text{squared"}\\ {a}^{3}\phantom{\rule{0.2em}{0ex}}\text{is read as}\phantom{\rule{0.2em}{0ex}}\text{"}a\phantom{\rule{0.2em}{0ex}}\text{cubed"}\end{array}\)

The table below lists some examples of expressions written in exponential notation.

Exponential Notation	In Words
\({7}^{2}\)	\(7\) to the second power, or \(7\) squared
\({5}^{3}\)	\(5\) to the third power, or \(5\) cubed
\({9}^{4}\)	\(9\) to the fourth power
\({12}^{5}\)	\(12\) to the fifth power

Example

Write each expression in exponential form:

\(\phantom{\rule{0.2em}{0ex}}16·16·16·16·16·16·16\)
\(\phantom{\rule{0.2em}{0ex}}\text{9}·\text{9}·\text{9}·\text{9}·\text{9}\)
\(\phantom{\rule{0.2em}{0ex}}x·x·x·x\)
\(\phantom{\rule{0.2em}{0ex}}a·a·a·a·a·a·a·a\)

Solution

The base 16 is a factor 7 times.	\({16}^{7}\)
The base 9 is a factor 5 times.	\({9}^{5}\)
The base \(x\) is a factor 4 times.	\({x}^{4}\)
The base \(a\) is a factor 8 times.	\({a}^{8}\)

Example

Write each exponential expression in expanded form:

\(\phantom{\rule{0.2em}{0ex}}{8}^{6}\phantom{\rule{0.2em}{0ex}}\)
\(\phantom{\rule{0.2em}{0ex}}{x}^{5}\)

Solution

The base is \(8\) and the exponent is \(6,\) so \({8}^{6}\) means \(8·8·8·8·8·8\)

The base is \(x\) and the exponent is \(5,\) so \({x}^{5}\) means \(x·x·x·x·x\)

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

Example

Simplify: \({3}^{4}.\)

Solution

	\({3}^{4}\)
Expand the expression.	\(3\cdot 3\cdot 3\cdot 3\)
Multiply left to right.	\(9\cdot 3\cdot 3\)
	\(27\cdot 3\)
Multiply.	\(81\)

Simplifying Expressions with Exponents