# Integer Exponents

An expression like \$2^3\$ means “2 multiplied by itself 3 times.” Here 2 is called a base, 3 is called an exponent, and the entire expression \$2^3\$ is read as “the 3rd power of 2,” or “2 to the 3rd power,” or sometimes just “2 to the 3rd.” In this lesson, you will learn some fundamental rules for working with this kind of expression, including when the exponent is zero or a negative integer.

## Computing powers

An expression like \$2^3\$ means “2 multiplied by itself 3 times.” That is,

\$\$2^3 = 2 ⋅ 2 ⋅ 2 = 4 ⋅ 2 = 8\$\$

(In this lesson we will use a dot to represent multiplication instead of parentheses, so \$2 ⋅ 3\$ means “2 times 3.”)

Compute the values of the following expressions which involve exponents.

## Exponents and multiplication

Let’s look at what can happen when you mix multiplication with using exponents.

We’ll start by examining the product \$2^2 ⋅ 2^3\$. By expanding \$2^2\$ and \$2^3\$, you can see that

\$\$ {\table 2^2 ⋅ 2^3, =, (2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2); , =, 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 } \$\$
 What is the total number of 2s being multiplied together?
 Write the expression above using only one exponent.
 What are the values of \$2^2\$ and \$2^3\$, and their product?
 Does this product match the value you found for \$2^2 ⋅ 2^3\$ by combining the two exponents?

Each row of the table below has an expression involving multiple exponents. Combine it into an expression involving only one exponent.

 Rewrite the expression \$6^10 ⋅ 6^20\$ in this way.

To multiply two exponential expressions with the same base together, you add the exponents. That is:

\$a^c ⋅ a^d = a^{c+d}\$ for any positive integers \$c\$ and \$d\$.

## Zero and negative exponents

Let’s try to figure out what \$2^0\$ should be. If we want to assign a value to it that follows the rule from expMultQn1, then

\$\$ 2^0 ⋅ 2 = 2^0 ⋅ 2^1 = 2^{0+1} = 2^1 = 2 \$\$
 This gives us the equation \$2^0 ⋅ 2 = 2\$. If we solve this equation for \$2^0\$ (by dividing both sides by \$2\$), what is the result?

This argument will work for any number \$a\$, as long as you’re allowed to divide by \$a\$; that is, as long as \$a ≠ 0\$. Because of this, we can say that:

Whenever \$a\$ is nonzero, \$a^0=1\$.

Now let’s see if we can figure out what powers with negative exponents should be. We’ll start by looking at \$3^{-1}\$. If we want to assign a value to it that follows the rule from expMultQn1, then what must be true about the product \$3^{-1} ⋅ 3\$?

This gives an equation for \$3^{-1}\$ (namely, \$3^{-1} ⋅ 3 = 1\$). What is the result of dividing both sides of this equation by \$3\$?

Now we’ll look at \$3^{-2}\$. What does the rule from expMultQn1 say about the product \$3^{-2} ⋅ 3^2\$?

Using the fact that \$3^2=9\$, what equation does this give us for \$3^{-2}\$?

What is the result of dividing both sides of this equation by \$9\$?

You can do this kind of calculation whenever you raise a nonzero number to a negative power. Because of this, we can say that:

If \$a ≠ 0\$ and \$d\$ is a positive integer, then \$\$a^{-d}=1/a^d\$\$.

Then if \$a ≠ 0\$, \$a^c ⋅ a^d = a^{c+d}\$ for any integers \$c\$ and \$d\$, even if \$c\$ or \$d\$ is not positive. Also, if \$a ≠ 0\$ then \$\$a^{-d}=1/a^d\$\$ for any integer \$d\$, even if \$d\$ is not positive.

Using these definitions, fill in the following table for \$2^d\$. Your answers will be plotted on the grid to the left.

\$d\$\$2^d\$\$y=2^d\$

Now, we’ll look at the expression \$(2 ⋅ 4)^3\$, which has multiplication inside an exponent. As above, this is equal to

\$\$ (2 ⋅ 4) ⋅ (2 ⋅ 4) ⋅ (2 ⋅ 4) \$\$

which can be rearranged into

\$\$ (2 ⋅ 2 ⋅ 2) ⋅ (4 ⋅ 4 ⋅ 4) \$\$
 What is the total number of 2s being multiplied together?
 What is the total number of 4s being multiplied together?

Rewrite the expression \$(2 ⋅ 4)^3\$ as shown, and then compute its value.

 Using a calculator if necessary, compute \$(2 ⋅ 4)^3\$ by first multiplying and then applying the exponent.
 Are the results of these two computations the same?

Each row of the table below has an expression that is a power of a product. Split it into an expression that is the product of two simpler powers.

 Rewrite the expression \$(2 ⋅ 3)^50\$ in this way.

In general:

\$(a ⋅ b)^d = a^d ⋅ b^d\$ for any integer \$d\$.

Now, we’ll look at what happens when you take an expression that already has an exponent in it and raise it to another power. As an initial example, take

\$\$ (4^2)^3 \$\$

This can be expanded into

\$\$(4 ⋅ 4) ⋅ (4 ⋅ 4) ⋅ (4 ⋅ 4)\$\$
 What power of 4 is \$(4^2)^3\$ equal to? (As before, count the number of 4s in the expanded expression.)
 Compute \$(4^2)^3\$ directly.
 Are the results of these two computations the same?

Write each expression in the table below using only one exponent.

 What power of \$6\$ is \$(6^5)^10\$ equal to?

In general:

\$(a^c)^d = a^{cd}\$ for any integers \$c\$ and \$d\$.

Let’s say we want to compute \$\$(3/4)^{-1}\$\$. It should be equal to \$\$1/(3/4)\$\$. Because

\$\$ 3/4 ⋅ 4/3 = 12/12 = 1 \$\$

we know that \$\$(3/4)^{-1}=1/(3/4)=4/3\$\$.

 What is \$\$(5/4)^{-1}\$\$?

As you can see:

For any fraction \$\$a/b\$\$ with \$a ≠ 0\$ and \$b ≠ 0\$, \$\$(a/b)^{-1}=b/a\$\$.

You can use this fact along with the rule for \$(a^c)^d\$ to find negative powers of fractions. For example, we can write

\$\$ (2/3)^{-2}=[(2/3)^{-1}]^2 \$\$
 What is \$(2/3)^{-1}\$?
 What is the square of that number?
 What is \$(2/3)^{-2}\$?

In this way, you can see that:

For any fraction \$\$a/b\$\$ with \$a ≠ 0\$ and \$b ≠ 0\$ and any integer \$n\$, \$\$(a/b)^{-n}=b^n/a^n\$\$.

Fill in the following table for \$(2/3)^d\$. Your answers will be plotted on the grid to the left. (Use a calculator.)

\$d\$\$\$(2/3)^d\$\$ \$\$y=(2/3)^d\$\$
(round to three
decimal places)

Use the rules from the previous questions to simplify the expressions in the table below.