Exponents

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Activity for this section:

Properties of Powers, Integer Powers

Exponents

Recall that multiplication can be used to simplify addition. For example, if we have 3 groups with 5 objects in each group, when we have “5 objects” 3 times or

5 + 5+ 5 = 15.

We simplify this idea by saying that there are

3 ×5 = 15

total objects.

We can do a similar technique to simplify multiplication. First, consider the expression

5 × 5× 5.

Which of the following questions could be answered by calculating $5 \times 5 \times 5$ ?

I have 5 trout, 5 carp, and 5 tuna. How many total fish do I have? I wrote 125 letters and put 5 letters in each envelope. How many envelopes did I use? I have 5 crates. Each crate contains 5 boxes of markers, and each box contains 5 markers. How many total markers do I have? None of these.

You may remember that a shorter way of writing $5 \times 5 \times 5$ is by writing $5^3$ read as “5 to the power of 3” or “five to the third”.

For counting numbers $A$ and $B$ , the expression $A^B$ is equal to

A◟-×-A-×◝◜⋅⋅⋅×A◞. B copies

We call $A$ the base and $B$ the power or exponent.

Evaluate $5^3$ .

$\answer {125}$

Rewrite $7^4$ as a product.

$\answer {7} \times \answer {7} \times \answer {7} \times \answer {7}$

Adding powers

Sometimes, we see addition inside a power. For example, we might see $4^{2+3}$ . Where does this come from? Let’s explore!

Let’s understand the meaning of $4^2\times 4^3$ . Well, $4^2$ is equal to $\answer [given]{16}$ and $4^3$ is equal to $\answer [given]{64}$ . This means that $(4^2)\times (4^3)$ represents the total number of objects in $\answer [given]{16}$ groups with $\answer [given]{64}$ objects in each group.

As we saw above, $4^2$ is also equal to $\answer {4} \times \answer {4}$ , and $4^3$ is equal to $\answer {4} \times \answer {4} \times \answer {4}$ . Using the associative commutative distributive property of multiplication, we see that

2 3 4 × 4 = (4 × 4)× (4 × 4× 4) = 4 × 4× 4× 4× 4.

This last product is equal to $4^{\answer {5}}$ . We ended up with 5 “4”s on the right hand side because we put together 2 “4”s and 3 “4”s. That is, we added subtracted multiplied divided 2 and 3 to find the total number of “4”s we have. In the end, we see that

2 3 2+3 2− 3 2×3 2÷ 3 4 × 4 = 4 .

For counting numbers $A$ , $B$ , and $C$ , the expression $A^{B+C}$ is equal to $A^B \times A^C.$

Rewrite $7^{3+5}$ as a product.

$7^{\answer {3}}\times 7^{\answer {5}}$

Rewrite $6^2 \times 6^7$ as a single base to one power. (Do not include “ $+$ ” in your answer!)

$\answer {6}^{\answer {9}}$

Multiplying Powers

We’ve discussed addition inside a power. What about multiplication? Why might we end up with $4^{3 \times 2}$ ? Where does this come from? Let’s explore!

Let’s understand the meaning of $(4^2)^3$ . Again, $4^2$ is equal to $\answer {4} \times \answer {4}$ . Raising $4^2$ to the power of 3 means multiplying together $3$ copies of $4^2$ . That is, $(4^2)^3$ is equal to

(42)3 = (4× 4)3 = (4× 4)× (4× 4)× (4× 4).

According to the grouping pictured here, we have $\answer {3}$ groups with $\answer {2}$ “4”s in each group. In other words, we have $3+2$ $3-2$ $3\times 2$ $3\div 2$ total “4”s multiplied together!

For counting numbers $A$ , $B$ , and $C$ , the expression $A^{B\times C}$ is equal to $(A^C)^B$ or $(A^B)^C$ .

Which of the following is equal to $7^{3\times 5}$ ?

$7^{3+5}$ $(7^5)^3$ $\frac {7^3}{7^5}$ $7^3 \times 7^5$

Rewrite $(6^2)^7$ using a single power. (Do not include “ $\times$ ” in your answer!)

$6^{\answer {14}}$

Power of 0

What if the number in the exponent is 0? For example, what is the meaning of the expression $4^0$ ? We’d like the “multiplication to addition” property of exponents (like the one in Definition 9) to always be true. If that’s true, then let’s explore the meaning behind $4^0$ .

If the “multiplication to addition” property of exponents is true, what should $4^3 \times 4^0$ be equal to? $4^{\answer {3}}$

In other words, when we multiply a number by $4^0$ , we should just just end up with the number we started with. What special number can we multiply a starting number by so that it doesn’t change? $\answer {1}$

For a counting number $A$ ,

A0 = 1.

Integer powers

We’ve explored addition and multiplication within an exponent. What about subtraction? In particular, let’s give meaning to an expression like “ $5^{-2}$ ”. Again, we’d like the “multiplication to addition” property of exponents (like the one in Definition 9) to always be true.

If the “multiplication to addition” property of exponents is true, what should $5^2 \times 5^{-2}$ be equal to? $5^{\answer {0}}$ And what single number is this expression equal to? $\answer {1}$

In other words, when we multiply $5^2$ by $5^{-2}$ , we should end up with 1. Note that $5^2 = 25$ . Combining these two facts, how should we define $5^{-2}$ ?

$5^{-2} = \frac {1}{\answer {25}}$

For counting numbers $A$ and $B$ ,

A −B = -1B. A

Rewrite $4^{-3}$ as a fraction without exponents.

$\frac {1}{\answer {64}}$

Rewrite $6^{-1}$ as a fraction without exponents.

$\frac {1}{\answer {6}}$

Rewrite $\frac {1}{36}$ using a negative exponent.

$6^{\answer {-2}}$

2025-08-08 20:56:33