Properties of exponents

In this chapter, you’ll learn what exponents mean and how to use them in expressions and algebraic equations.

Understanding exponents

Exponents are the small numbers written up and to the right of a number or variable (for example, $2^4$, $x^2$, $10^{-3}$). An exponent tells you how many times to multiply a number by itself. This is also called raising a number to a power.

For example, in $2^4$, the exponent is $4$, so you multiply four factors of $2$:

$2^4=1\ast 2\ast 2\ast 2\ast 2=16$

See below for a more detailed breakdown:

$10^4$ has an exponent of $4$, so we call it “ten to the fourth power.” You multiply four factors of $10$:

$10\ast 10\ast 10\ast 10=10,000$

$3^2$ has an exponent of $2$, so we call it “three to the second power.” You multiply two factors of $3$:

$3\ast 3=9$

A number to the second power is often referred to as “squared.” In this case, it’s more accurate to say “three squared.”

Understanding negative exponents

Negative exponents work like positive exponents, but they create a reciprocal. Instead of multiplying by the base, you divide by the base the number of times indicated by the exponent.

For example, in $2^{-4}$, the exponent is $-4$, so you divide by $2$ four times:

$$1/(2\ast 2\ast 2\ast 2)=1/16=0.0625$$

See below for a more detailed breakdown:

A shortcut is to rewrite a negative exponent by moving the base across the fraction bar:

$$2^{-2}=1/(2^2)=1/4$$

Or, moving it back the other way:

$$1/(2^{-2})=2^2=4$$

You can think of a negative exponent as a way to move a factor from the numerator to the denominator (or from the denominator to the numerator).

Exponents on parentheses

There are two different situations involving exponents on parentheses:

• The expression inside the parentheses does not include addition/subtraction.
• The expression inside the parentheses does include addition/subtraction.

For the first situation (no addition/subtraction inside the parentheses), apply the exponent to each factor inside. In the example below, the exponent $2$ applies to both $3$ and $x$:

$$(3x{)}^2=3^2\ast x^2$$

For the second situation (addition/subtraction inside the parentheses), you must expand the expression. For a squared binomial, this means FOIL. If you’re unfamiliar with FOIL, see the chapter Simplifying Expressions

To see why expansion is needed, rewrite the exponent as repeated multiplication:

$$(x+2{)}^2=(x+2)(x+2)$$

Because the expression is multiplied by itself, you FOIL:

$$(x+2)(x+2)=x^2+4x+4$$

Exponents on fractions

When the entire fraction is raised to an exponent, both the numerator and denominator are raised to that exponent:

$$(\frac{2x}{(x+1)}{)}^2=\frac{(2x{)}^2}{(x+1{)}^2}$$

After that, continue simplifying using the rules discussed in the chapter “Rules of simplification”. To finish the example above, distribute the exponent in the numerator and FOIL the denominator:

$$\frac{4x^2}{(x^2+2x+1)}$$

Exponents in arithmetic

When simplifying expressions, follow the order of operations (PEMDAS). Exponents come after parentheses and before multiplication/division.

When you reach an exponent while simplifying, evaluate it before moving on. This makes the rest of the expression easier to work with.

Try an example. When you reach the exponent step in the order of operations, simplify the exponent before continuing.

Simplify the following expression:

$$2+3(4^2-2(3))$$

Exponents in algebra

Multiplying and dividing with exponents

When multiplying variables with exponents, first multiply any numbers in front as usual. Then add the exponents on like bases. Remember: if a variable has no written exponent, its exponent is $1$.

$$x^2\ast x^3=x^{2+3}=x^5$$

Division works similarly. Divide any numbers in front first. Then subtract exponents on like bases (top exponent minus bottom exponent):

$$\frac{x^3}{x^2}=x^{3-2}=x^1=x$$

Try some examples:

What is $x\ast x^3$ simplified?

What is $2x^2\ast 3x^3$ simplified?

What is $4x^2/2x$ simplified?

Exponent raised to an exponent (double exponents)

Sometimes an exponent is raised to another exponent, like this:

$$(x^2{)}^2$$

When this happens, start with the exponent that’s deepest in the expression and work outward. In this example, evaluate $2^2$ first to get $4$, so the expression simplifies to $x^4$. Simplifying first also makes it easier to use the expression in equations.

Another way to think about this is that the two exponents multiply. This matters because exponents aren’t always just numbers; they can be variables. For example, if we have $(2^{3x}{)}^{2y}$, we multiply the exponents to combine them:

$2^{3x\ast 2y}=2^{6xy}$

Adding and subtracting with exponents

You may have seen an equation like this:

$$x^3+x^2+x+1=0$$

The key is that this equation is already in its simplified form because you may only add or subtract terms with the same variable and the same exponent.

That means you can’t simplify $x^3+x^2$ any further. Treat terms with different exponents as different kinds of terms. However, if the exponents match, you can combine them:

$$x^3+2x^3=3x^3$$

You’ve already followed this rule when adding ordinary numbers, since each number can be thought of as having an exponent of $1$:

$$4+6-3=4^1+6^1-3^1=7$$

Reversing exponents

For details on how to reverse exponents in algebra (that is, how to solve for $x$ when $x$ has an exponent), see the chapter Radicals and rational exponents.

Key points

Power. A number “to a power of…” means that it has that “power” as an exponent. This exponent indicates how many times we multiply $1$ by our original number.

Square. A number is squared when it is to the second power (exponent of $2$).

Multiplying/dividing. Multiply or divide any numbers in front of the variables, then multiply or divide the variables. This is when the exponent rule will come into play: add exponents when multiplying, and subtract them when dividing.

Adding/subtracting. Add or subtract variables.

Double exponents. Think of PEMDAS, and take care of the exponent that is deepest into the term first. Find the value of the exponent raised to another exponent, and that will be the new exponent of the actual number.