A Product Of Repeated Factors

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

A repeated addition tin be written equally a multiplication. So, if we get 10 letters one day, 10 the side by side day, and ten the day after, nosotros take got, in full, messages.

The repeated addition of the number x is written equally a multiplication of x past 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, at that place is a way to write repeated multiplication. For instance, allow usa suppose that someone receives a picture on social media. This person shares it in the start infinitesimal to four of his or her friends. In the following minute, each of the four friends shares it to another 4 people. And, in the 3rd minute, each of the people who received the movie in the previous minute shares information technology to another four people. The number of people who got this picture in the third minute is

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

The number is chosen a ability of 4. The gene 4 that is repeated is chosen the base, and the exponent 3 is the number of times the gene appears in the repeated multiplication. (We could also say it is ane more than the number of times 4 is multiplied past itself.)

Let united states of america summarize what we have just learned about powers.

Definition: Powers and Exponential Form

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

The base of operations is the cistron 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, nosotros say it is written in exponential course.

When a ability is written as a repeated multiplication, we say information technology is written in expanded form.

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

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

The identity property multiplication tells us that multiplying any number past one does non change this number. Information technology follows that all the powers of 1 are just 1: , , , , and so on.

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

Allow united states of america get through several examples to check and deepen our understanding.

Example 1: Writing a Repeated Multiplication in Exponential Form

Write in exponential form.

Respond

The expression is a repeated multiplication of the gene 7. It tin be written in exponential course, that is, in the form , where is the factor that is repeatedly multiplied past itself, here 7, and is the number of times this cistron appears in the repeated multiplication, hither 4. Hence,

Example 2: Writing a Power in Expanded Course

Write in expanded class.

Respond

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

Hence,

Instance 3: Writing a Power from Its Name

Write vii to the fourth power using digits.

Answer

Nosotros are asked to write 7 to the fourth power. Recollect that a ability is written in the form . It is a shorthand for a repeated multiplication of a factor , with being the number of times this gene appears in the repeated multiplication. There are unlike ways to call : raised to the th power or to the ability of , or to the th power. Here, we have seven to the 4th power, so we observe that and . Hence, it is written every bit .

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

Express as a product of the aforementioned factor, and so discover its value.

Answer

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

The exponential class 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 class is

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

Permit u.s.a. await now at an instance of multiplications of powers of different bases.

Example v: Expanding a Multiplication of 2 Powers

Which of the following is equivalent to ?

Answer

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

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

Let us expect at another example to understand what happens when two powers of the same base of operations are multiplied. In this case, we have a ability of a variable, , instead of a number, and we volition see that it does not alter anything to the style we handle powers.

Example half-dozen: Multiplying Powers of the Same Base

Simplify .

Answer

We want to express as a single power. To visualize ameliorate how this expression can exist simplified, let u.s. first rewrite it by expanding both powers. We notice that our expression is

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

Nosotros 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 Dominion

The product of two powers that have the same base of operations is a power of this same base of operations 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.

Case 7: Rewriting Expressions Using Exponents

Which of the following expressions is equivalent to ?

Answer

The expression is a multiplication involving different factors. Therefore, it cannot be expressed every bit a single ability. However, we find that the factors 7 and 5 are repeated. We are going to use the commutative holding of multiplication to first rewrite our expression then 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 cistron , with existence the number of times this factor appears in the repeated multiplication. Hence, this expression is equivalent to

Using commutativity once again, we tin modify the guild of the factors and write it as

Example 8: Finding Equivalent Expressions for Repeated Multiplications

Which of the following expressions is equivalent to ?

Respond

The expression is a multiplication involving different factors. Therefore, information technology cannot exist expressed as a unmarried power. Nevertheless, we discover that the factors 2 and v are repeated. Using the commutative property of multiplication, nosotros tin can rewrite our expression by grouping the twos together and the fives together. We get

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

However using the commutative property of multiplication, nosotros can besides rewrite our starting time expression as which, using the associative belongings this time, is equivalent to

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

Hence, our expression is also equivalent to which can exist written every bit the fourth power of x: .

When nosotros evaluate , we observe it is 10‎ ‎000.

In conclusion, we accept institute that

Therefore, the right answer is option A: .

In this last example, note that in one case you have understood the method, y'all do not need to group all the identical factors. You tin 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 accept

This tin be summarized in this way.

Power of a Product

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

Example 9: Expressing a Repeated Multiplication of a Fraction with a Power

What is ?

Respond

An exponent represents the number of times a rational number is multiplied by itself.

In this expression, is multiplied by itself 7 times. So,

Key Points

Powers are numbers resulting from a repeated multiplication of a factor. Their full general form is where is chosen the base and the exponent, which means 's multiplied together. So, for instance, we have
When a number is written equally a power, we say it is written in exponential form. When a power is written every bit a repeated multiplication, we say it is written in expanded form.
The product of ii powers that have the same base of operations is a ability of this same base with an exponent equal to the sum of the exponents:
A given power of a product of factors is the product of each cistron raised to that given power: