Introduction to exponents

For $$2^3$$, three is the exponent on two. Similarly for $$x^7$$, seven is the exponent on $$x$$.

Exponents are shorthand for repeated multiplication. They tell us how many times a number and/or variable are multiplied together. For example, using an exponent, the multiplication problem $$3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$$ can be written as $$3^6$$. The exponent, six, in this example, tells us that we have six 3s multiplied together.

For example:


 * $$5 \cdot 5 \cdot 5 = 5^3$$

In the problem, $$2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$$, we can use an exponent to represent the count for the number of 2s that are multiplied together and an exponent to represent the count of the number of 5s that are multiplied together. This multiplication problem can be written with exponents as follows: $$2^3 \cdot 5^5$$. With the exponents, we can quickly see that we have three 2s and five 5s multiplied together.

Exponents are used in the same manner with variables.

Some examples
Let's take a look at some examples to see the variety of ways using exponents helps to simplify an expression.

Example 1:
$$7 \cdot 7 \cdot x \cdot x \cdot x \cdot x = 7^2 \cdot x^4$$, the exponents tell us that we have 2 7s and 4 x's multiplied together.

Example 2:
$$8 \cdot 8 \cdot 8 \cdot x \cdot x \cdot y \cdot y \cdot y = 8^3 \cdot x^2 \cdot y^3$$, three 8s, two x's, and three y's are multiplied together.

Example 3:
In the expression $$4 \cdot a \cdot a \cdot b \cdot b \cdot b$$ we have one 4 which could be written as $$4^1$$ but typically, we write anything with an exponent of one without the exponent. So, one 4, two a's and three b's all multiplied together can be shortened to:


 * $$4 \cdot a^2 \cdot b^3$$

Remember that $$a \cdot b$$ is the same as $$a b$$ and that $$2 x y$$ is the same as $$2 \cdot x \cdot y$$. So, the expression above can also be written as:


 * $$4 a^2 b^3$$

Example 4:
In the expression $$5^3 m^4 n$$ the exponents tell us that we have three 5s, four m's and one n all multiplied together. To write this problem out the long way, we get


 * $$5^3 m^4 n = 5 \cdot 5 \cdot 5 \cdot m \cdot m \cdot m \cdot m \cdot n$$

Example 5:

 * $$9 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot z \cdot z \cdot z \cdot z \cdot z \cdot z \cdot z \cdot z \cdot z = 9 x^5 y z^9$$

Clearly, the shorthand method of using exponents is better than writing it out the long way.

Further simplifications
Now, let's look at ways to further simplify an expression.

Example 6:
Given $$2^3 = 2 \cdot 2 \cdot 2 = 8$$, then we can further simplify:


 * $$2^3 x^4 y^7 = 8 x^4 y^7$$

Example 7:
$$8^2 m^5 = 64 m^5$$

Example 8:
$$5^3 x^2 x^3 = 125 x^5$$, note that in the original there are 2 x's and 3 x's which makes 5 x's total.

Example 9:
$$4^3 \cdot a \cdot a \cdot a \cdot a \cdot a \cdot b \cdot b \cdot b \cdot b = 64 a^5 b^4$$

Negative exponents
A positive exponent signifies how many times to multiply by the number. A negative exponent signifies the opposite: how many times to divide by the number. This can easily be accomplished by moving the number and it's exponent to the denominator of a fraction.

Example 10:

 * $$m^{-5} = \frac{1}{m^5}$$

A harder problem
You can use your understanding of exponents, that they tell you how many times to multiply or divide a number, to simplify expressions which have one or more variables combined in various ways.

Example 11:
Take a careful look at this fraction with exponents: $$\frac{(r^4 v^5)^4}{(r^{-1} v^{-4})^{-5}}$$. Let's simplify it one step at a time.


 * $$\frac{(r^4 v^5)^4}{(r^{-1} v^{-4})^{-5}} =$$
 * $$ = \frac{(r^{16} v^{20})}{(r^5 v^{20})}$$, the most outer exponent tells us how many times the inner exponents are multiplied, so multiply the outer exponent by each of the inner exponents; remember that multiplying two negative numbers results in a positive.
 * $$ = \frac{r^{16}}{r^5}$$, there are the same number of v's in the numerator and the denominator, so these cancel out, leaving only the r's.
 * $$ = r^{11}$$, the 5 r's in the denominator cancel with 5 of the 16 r's in the numerator, leaving 11 r's in the numerator.