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When a number is added to itself many times, the result can be written as a multiplication expression. The aim of this lesson is to examine the case where a number is multiplied by itself many times.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

How to Write Many Multiplications

When a specific number is added to itself multiple times, the result is the same as multiplication.
Different numbers added by itself a random number of times.
But what if, instead, the number is multiplied by itself a certain number of times?
Different numbers multiplied by itself a random number of times.
Is there a short way to write this kind of multiplication expression? This will be explored in this collection.
Discussion

Powers, Exponents, and Bases

A power is the product of a repeated factor. A power expression consists of two parts. The base is the repeated factor and the exponent indicates how many times the base is used as a factor. Consider, for example, the power expression with base and exponent

A power expression 7^4 where 7 is the base and 4 is the exponent
In this example, is multiplied by itself times.
Most powers are read in the same way.
Expression Example Example
to the second power squared
to the third power cubed
raised to the power of raised to the fourth power
When a number is raised to the power of or the expression can be read as squared or cubed, respectively. Expressions for greater powers are all usually read as the last example on the table.
Pop Quiz

Rewriting Products

Rewrite the product of repeated factors as a power. Consider that to write the input should be

Random generator of products of repeated factors.
Example

Bonus Points in a Trivia Contest

Tearrik is participating on a trivia contest at school.

Tearrik raising his hand to answer a question

At the moment, he is on the bonus question round. Each question that he answers correctly doubles the amount of bonus points he gets. The first question is worth two bonus points. The table below displays the bonus points Tearrik will earn if he answers or questions correctly.

Questions Answered Correctly Points Simplify
a Select the power that represents how many points Tearrik will win if he answers three questions correctly.
b If Tearrik answers four questions correctly, he will receive bonus points. Rewrite this power as a product of repeated factors.
c Tearrik manages to answer five questions correctly before the bonus round ends. How many bonus points does he win?

Hint
a How many times is the number multiplied by itself?
b What does the exponent indicate?
c Do the smaller multiplications one at a time.

Solution
a Looking at the table, answering three questions correctly means the number is multiplied by itself times.
To write this product as power, it is important to identify the base and the exponent. The base is the number being multiplied and the exponent is the number of times that the base is multiplied.
Now that the base and the exponent have been identified, the expression can be written.
b First, look at the given power expression for the bonus points earned if Tearrik answers four questions correctly.
Looking at the expression, it is possible to determine which is the base and which is the exponent of the expression.
The base is the number being multiplied and the exponent is the number of times that the base is multiplied by itself. Knowing this, it is possible to write the given power as the product of repeated factors.
c It was mentioned previously that if Tearrik answers three or four questions correctly, he would get or points, respectively. With this in mind, it is possible to write a power expression for the total number of bonus points Tearrik receives after answering five questions correctly.
In this expression, the number is the number being multiplied and is the number of times that is multiplied by itself. Use this information to write the power expression as the product of repeated factors.
The resulting product of multiplying by itself times can now be determined. It might be helpful to start by calculating smaller products.
AssociativePropMult

Associative Property of Multiplication

Multiply

Multiply

AssociativePropMult

Associative Property of Multiplication

Multiply

Multiply

Multiply

Multiply

Therefore, Tearrik gains bonus points in the bonus question round if he answers questions correctly.


Pop Quiz

Evaluating Powers

Find the value of the given power.

Random generator of power expressions.
Discussion

Raising a Number to the Power of

The numbers that result from raising an integer to the power of appear frequently in math. These numbers are called perfect squares.

Concept

Perfect Square

A perfect square is a number that can be expressed as the square of an integer.
Example Rewrite as a Product Perfect Square? Explanation
Yes ${\color{#009600}{\bm{\Large{\checkmark }}}}$ is an integer.
No ${\color{#FF0000}{\bm{\Large{\times }}}}$ is not an integer.
No ${\color{#FF0000}{\bm{\Large{\times }}}}$ is not an integer.
Yes ${\color{#009600}{\bm{\Large{\checkmark }}}}$ is an integer.
Discussion

Raising to the Power of

Similar to perfect squares, there are numbers called perfect cubes.

Concept

Perfect Cube

A perfect cube is a number that can be expressed as the cube of an integer.
Example Rewrite as a Product Perfect Cube? Explanation
Yes ${\color{#009600}{\bm{\Large{\checkmark }}}}$ is an integer.
No ${\color{#FF0000}{\bm{\Large{\times }}}}$ is not an integer.
Yes ${\color{#009600}{\bm{\Large{\checkmark }}}}$ is an integer.
No ${\color{#FF0000}{\bm{\Large{\times }}}}$ is not an integer.
Example

Testing for Perfect Squares and Perfect Cubes

One of the questions that Tearrik could not answer in the trivia contest was about perfect squares. Now he wants to study harder so that he does not make the same mistake twice!

Tearrik_Studying

Determined to improve, he decides to also study perfect cubes. Help Tearrik solve the following exercises.

a Which of the following numbers are perfect squares? Select all that apply.
b Which of the following numbers are perfect cubes? Select all that apply.

Hint
a Note that the numbers between the squares of two consecutive integers cannot be perfect squares.
b Note that the numbers between the cubes of two consecutive integers cannot be perfect cubes.

Solution
a A perfect square is a number that can be expressed as the square of an integers. For example, the number can be written as the square of
Since is an integer, is a perfect square. To identify the numbers that are not perfect squares, consider the squares of and
Since and are integers, and are perfect squares. There are no integers between and This means that there are no perfect squares between and
A number line from 23 to 37 with increments of 1. Two points are plotted at 25 and 36. Values between these points are not perfect squares.

It can be noted that is not a perfect square. Now, to determine if is a perfect square, remember that the square of is The consecutive perfect squares are displayed in the table below.

Number Square Less Than, Greater Than, or Equal to

As shown in the table, is greater than but less than Since lies between two consecutive perfect squares, cannot be a perfect square. More perfect squares will be identified to determine if is a perfect square.

Number Square Less Than, Greater Than, or Equal to

The number is the square of Since is an integer, is a perfect square. The remaining number is The square of is The square of is Since is closer to than to it might be convenient to start with and explore the next perfect squares in decreasing order.

Number Square Less Than, Greater Than, or Equal to

Notice that is a perfect square. Now that all numbers have been examined, the results can be summarized in another table.

Number Perfect Square?
Yes
No
No
Yes
Yes
b A perfect cube is a number that can be expressed as the cube of an integer. For example, the cube of is Therefore, is a perfect cube.
Since all the given numbers are less than any perfect cubes among them can be expressed as the cube of an integer from to Make a table to write these cubes and look for the given numbers, starting with
Number Cube Less Than, Greater Than, or Equal to

Because can be expressed as the cube of the number it is a perfect cube. Using a similar reasoning, the number will be examined.

Number Cube Less Than, Greater Than, or Equal to

As shown, is a perfect cube. Now continue the table to see if is a perfect cube.

Number Cube Less Than, Greater Than, or Equal to

The number lies between two consecutive perfect cubes, so is not a perfect cube. Now the number will be tested. It is a good thing that is really close to

Number Cube Less Than, Greater Than, or Equal to

The number lies between the perfect cubes and meaning that is not a perfect cube. Finally, will be tested.

Number Cube Less Than, Greater Than, or Equal to

Every number was examined. The results are summarized in the table below.

Number Perfect Cube?
Yes
Yes
No
No
Yes

Notice that a number can be both a perfect cube and a perfect square!

Pop Quiz

Identifying Perfect Squares and Perfect Cubes

Determine whether the given number is a perfect square, a perfect cube, both, or neither.

Perfect square or perfect cube
Closure

Even and Odd Powers

One thing that is worth exploring is what happens to negative numbers when they are raised to some power. This is interesting because, for example, if is squared, the result is a positive number!
In general, the square of any number is positive. This can be extended to any case where the exponent is an even number, no matter if the base is a positive or negative number. The factors can be grouped in pairs, each of which results in a positive number. Consider another example,
Breaking down (-2) to the power of 6 as the multiplication of six (-2)s: (-2) * (-2) * (-2) * (-2) * (-2) * (-2).
But what happens if the exponent is an odd number? To answer this, first remember that if a positive number is multiplied by a negative number, the result is a negative number. If is multiplied by the product is a negative number.
Consider how exponents are written. Since is being multiplied by itself times, the expression can be written as This is an odd exponent.
When raising a negative number to an odd power, the result is always negative. This happens because grouping the factors in pairs always results with one factor that cannot be included in a pair, which changes the sign of the product. This leads to the following conclusion.
Exponent Result
Even Always positive
Odd Sign depends on the base


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