# Explainer: Powers and Exponents

In this explainer, we will learn how to identify the base and exponent in power formulas, write them in exponential, expanded, and word forms, and evaluate simple powers.

A repeated addition can be written as a multiplication. So, if we get 10 messages one day, 10 the next day, and 10 the day after, we have got, in total, messages.

The repeated addition of the number 10 is written as a multiplication of 10 by the number of times 10 appears in the repeated addition. (We could also say it is one more than the number of times 10 is added to itself.)

Similarly, there is a way to write repeated multiplication. For instance, let us suppose that someone receives a picture on social media. This person shares it in the first minute to four of his or her friends. In the following minute, each of the four friends shares it to another four people. And, in the third minute, each of the people who received the picture in the previous minute shares it to another four people. The number of people who got this picture in the third minute is

This repeated multiplication of a given factor (here 4) can be written as a power of this factor; namely,

The number is called a power of 4. The factor 4 that is repeated is called the base, and the exponent 3 is the number of times the factor appears in the repeated multiplication. (We could also say it is one more than the number of times 4 is multiplied by itself.)

Let us summarize what we have just learned about powers.

### Definition: Powers and Exponential Form

Powers are numbers resulting from a repeated multiplication of a factor. Their general form is where is called the base and the exponent.

The base is the factor repeatedly multiplied by itself, and the exponent is the number of times appears in the repeated multiplication.

When a number is written as a power, we say it is written in exponential form.

When a power is written as a repeated multiplication, we say it is written in expanded form.

You may have already learned about the square and cube of a number. Squaring is the same as raising to the second power; it is multiplying a number by itself: . A number raised to the third power, or the power of three, is cubed: .

Note that any number can be written as the first power of itself, .

The identity property multiplication tells us that multiplying any number by 1 does not change this number. It follows that all the powers of 1 are simply 1: , , , , and so on.

Also, raising any nonzero number to the zeroth power gives 1.

Let us go through several examples to check and deepen our understanding.

### Example 1: Writing a Repeated Multiplication in Exponential Form

Write in exponential form.

The expression is a repeated multiplication of the factor 7. It can be written in exponential form, that is, in the form , where is the factor that is repeatedly multiplied by itself, here 7, and is the number of times this factor appears in the repeated multiplication, here 4. Hence,

### Example 2: Writing a Power in Expanded Form

Write in expanded form.

A power is written in the form . It is a shorthand for a repeated multiplication of a factor , with being the number of times this factor appears in the repeated multiplication. Here, and .

Hence,

### Example 3: Writing a Power from Its Name

Write seven to the fourth power using digits.

We are asked to write seven to the fourth power. Recall that a power is written in the form . It is a shorthand for a repeated multiplication of a factor , with being the number of times this factor appears in the repeated multiplication. There are different ways to call : raised to the th power or to the power of , or to the th power. Here, we have seven to the fourth power, so we find that and . Hence, it is written as .

### Example 4: Writing a Power in Expanded Form and Evaluating it

Express as a product of the same factor, and then find its value.

The number is written in exponential form and we want first to write it in expanded form.

The exponential form is a shorthand for a repeated multiplication of a factor , with being the number of times this factor appears in the repeated multiplication. Hence, in expanded form is

Then, we need to evaluate this product. Using associativity, we can write

Let us look now at an example of multiplications of powers of different bases.

### Example 5: Expanding a Multiplication of Two Powers

Which of the following is equivalent to ?

The expression involves the multiplication of two powers, and . Let us expand each of them. Recall that a power, written in the form , is a shorthand for a repeated multiplication of a factor , with being the number of times this factor appears in the repeated multiplication. Hence, we have and

Now, we simply multiply them together to find an expression equivalent to :

Let us look at another example to understand what happens when two powers of the same base are multiplied. In this example, we have a power of a variable, , instead of a number, and we will see that it does not change anything to the way we handle powers.

### Example 6: Multiplying Powers of the Same Base

Simplify .

We want to express as a single power. To visualize better how this expression can be simplified, let us first rewrite it by expanding both powers. We find that our expression is

The parentheses can be removed here using the associative property of multiplication. This repeated multiplication of the factor involves in total . Hence, it can be written as .

We have seen in the previous two examples that when we multiply two powers of the same base, we get

This is known as the product rule.

### The Product Rule

The product of two powers that have the same base is a power of this same base with an exponent equal to the sum of the exponents:

In the following two examples, we are going to see how the commutative property of multiplication is used to rewrite expressions involving repeated multiplications of different factors.

### Example 7: Rewriting Expressions Using Exponents

Which of the following expressions is equivalent to ?

The expression is a multiplication involving different factors. Therefore, it cannot be expressed as a single power. However, we notice that the factors 7 and 5 are repeated. We are going to use the commutative property of multiplication to first rewrite our expression so that all identical factors are grouped together.

For instance, we may rewrite as

It is now easy to see that 7 is used twice, 3 once, and 5 thrice. Recall that a power, written in the form , is a shorthand for a repeated multiplication of a factor , with being the number of times this factor appears in the repeated multiplication. Hence, this expression is equivalent to

Using commutativity again, we can change the order of the factors and write it as

### Example 8: Finding Equivalent Expressions for Repeated Multiplications

Which of the following expressions is equivalent to ?

The expression is a multiplication involving different factors. Therefore, it cannot be expressed as a single power. However, we notice that the factors 2 and 5 are repeated. Using the commutative property of multiplication, we can rewrite our expression by grouping the twos together and the fives together. We get

Recall that a power, written in the form , is a shorthand for a repeated multiplication of a factor , with being the number of times this factor appears in the repeated multiplication. Hence, our expression is equivalent to

Still using the commutative property of multiplication, we can also rewrite our first expression as which, using the associative property this time, is equivalent to

We see that the number inside each pair of brackets is , that is, 10.

Hence, our expression is also equivalent to which can be written as the fourth power of 10: .

When we evaluate , we find it is 10,000.

In conclusion, we have found that

Therefore, the correct answer is option A: .

In this last example, note that once you have understood the method, you do not need to group all the identical factors. You can simply count the number of times each factor appears and then write an equivalent power to the repeated multiplication of each factor.

We have found in the previous two examples that when raising a product of two factors to the th power, we have

This can be summarized in this way.

### Power of a Product

A given power of a product of factors is the product of each factor raised to that given power:

What is ?