# The Laws of Exponents

We write \$a^d\$ to mean “\$a\$ multiplied by itself \$d\$ times.” Here \$a\$ is called a base, \$d\$ is called an exponent, and the entire expression \$a^d\$ is called “the \$d\$th power of \$a\$.” In this lesson, you will learn some rules for simplifying expressions involving exponents.

## Review — Computing powers

Compute the values of the following expressions which involve exponents. Note that in this lesson we will use a dot to represent multiplication instead of parentheses, so \$2 ⋅ 3\$ means “2 times 3.”

## 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

\$\$ 2^2 ⋅ 2^3 = (2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) \$\$
 What is the total number of 2s being multiplied together?
 Write \$2^2 ⋅ 2^3\$ 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 \$a^10 ⋅ a^20\$ in this way.

As you can see:

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

Now, we’ll look at the expression \$(2 ⋅ 4)^3\$. 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 \$(ab)^50\$ in this way.

In general:

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

## A power of a power

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 \$a\$ is \$(a^5)^10\$ equal to?

In general:

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

## The value of \$a^0\$

Using the rules you’ve just seen for how \$a^d\$ works when \$d\$ is a positive integer, we’d like to figure out how to assign a value to \$a^0\$.

Let’s look at \$2^0\$. If we want to assign a value to it that follows the rule \$a^c ⋅ a^d = a^{c+d}\$ from expMultQn1, then it must be true that

\$\$ 2^0 ⋅ 2^d = 2^{0+d} = 2^d \$\$

whenever \$d\$ is a positive integer.

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

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

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

We got this by using the rule in expMultQn1, but it also works well with the other two rules. For example, we’ll check that the rule \$(a ⋅ b)^d = a^d ⋅ b^d\$ from expMultQn2 still holds for \$d = 0\$.

 If neither \$a\$ nor \$b\$ is zero, is \$ab=0\$?
 If neither \$a\$ nor \$b\$ is zero, what is \$(ab)^0\$?
 If neither \$a\$ nor \$b\$ is zero, what is \$a^0 b^0\$?
 If neither \$a\$ nor \$b\$ is zero, does \$(ab)^0=a^0 b^0\$?

You can similarly check that the rule \$(a^c)^d = a^{cd}\$ from powerPowerQn still holds if \$c = 0\$ or \$d = 0\$.

## Negative exponents

Just as in zeroPower for \$a^0\$, we would now like to assign a value to \$a^c\$ where \$c\$ is a negative number, which follows the laws of exponents from the earlier questions.

We’ll start by looking at \$3^{-d}\$, where \$d\$ is a positive integer. If we want to assign a value to it that follows the rule \$a^c ⋅ a^d = a^{c+d}\$ from expMultQn1, then what must be true about the product \$3^{-d} ⋅ 3^d\$?

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

In the last question, you saw that we would like to define \$3^{-d}\$ to be \$\$1/3^d\$\$. The same logic tells us how to define \$a^{-d}\$ whenever \$a ≠ 0\$ (so we can divide by \$a^d\$):

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

This definition actually makes \$\$a^{-d}=1/a^d\$\$ for any integer \$d\$.

Using this definition, 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\$
 When \$d\$ changes from 0 to 1, \$2^d\$ changes from 1 to 2: that is, \$2^d\$ is multiplied by 2. Similarly, when d changes from 1 to 2, \$2^d\$ changes from 2 to 4: that is, it again is multiplied by 2. When d changes from 2 to 3, what does \$2^d\$ get multiplied by?
 When \$d\$ changes from \$-3\$ to \$-2\$, \$2^d\$ changes from \$\$1/8\$\$ to \$\$1/4\$\$: that is, it is multiplied by 2. When \$d\$ changes from \$-2\$ to \$-1\$, what does \$2^d\$ get multiplied by?
 In general, when \$d\$ increases by 1 in this table, what does \$2^d\$ get multiplied by?

We defined \$a^{-d}\$ so that it would satisfy the rule \$a^c a^d=a^{c+d}\$ when \$c = -d\$. In fact, using \$a^0 = 1\$ and \$\$a^{-d}=1/a^d\$\$ makes all three of our fundamental laws of exponents hold for any integer exponents, positive or zero or negative, as long as the bases of the powers are never zero.

We could check these rules for any specific \$c\$ and \$d\$ values, including negative ones, by expanding each side of each equation as a fraction, and counting the number of times \$a\$ and \$b\$ occurred in each numerator and denominator.

Whenever \$a\$ and \$b\$ are nonzero and \$c\$ and \$d\$ are integers, \$\$ a^c a^d = a^{c + d} \$\$ \$\$ (a b)^d = a^d b^d \$\$ \$\$ (a^c)^d = a^{cd} \$\$

Use these rules to simplify the expressions in the table below. You may assume that \$a\$ and \$b\$ are nonzero.

You can use the rule for negative exponents to find a rule for subtraction in exponents:

\$\$ a^{c - d} = a^{c + (-d)} = a^c a^{-d} = a^c ⋅ 1 / a^d = a^c / a^d \$\$

That is:

Whenever \$a\$ is nonzero and \$c\$ and \$d\$ are integers, \$\$a^{c - d} = a^c / a^d\$\$.

This subtraction rule gives you another way to compute \$a^0\$, by noticing that it is equal to \$a^{d-d}\$.

What do you get when you apply the subtraction rule to \$a^{d - d}\$?

The equation \$y=a^x\$ has been graphed to the left, with a slider for \$a\$. Notice that this graph consists of separated points instead of a curve, because we only know how to find \$a^x\$ when \$x\$ is an integer.

 Use the slider to set \$a\$ to some value which is larger than 1. As \$x\$ increases (goes from left to right), does \$a^x\$ always increase, always decrease, or do neither?
 Now set \$a\$ to some value between 0 and 1. As \$x\$ increases, does \$a^x\$ always increase, always decrease, or do neither?
 Finally, set \$a\$ to some negative value. As \$x\$ increases, does \$a^x\$ always increase, always decrease, or do neither?

Using the rule for addition of exponents, you can see that:

\$\$ a^{x+1}=a^x ⋅ a^1 = a^x ⋅ a \$\$
 When \$x\$ increases by 1, what does \$a^x\$ get multiplied by? (Hint: The answer is a letter, not a number.)