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1. Powers
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Powers

This lesson delves into the mathematical concepts of perfect squares, perfect cubes, and powers. It focuses on understanding the base and exponent in power expressions. The lesson is designed to help individuals like Tearrik, who wants to improve his understanding of these topics after failing to answer trivia questions about them. The concepts are explained through practical examples, such as calculating bonus points in a trivia game. Understanding these principles is crucial for various real-world applications, including geometry and advanced algebra.
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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 7 and exponent 4.

A power expression 7^4 where 7 is the base and 4 is the exponent

In this example, 7 is multiplied by itself 4 times. 7^4 = 7 * 7 * 7 * 7_4

Most powers are read in the same way.
Expression Example 1 Example 2
2^2 2 to the second power 2 squared
7^3 7 to the third power 7 cubed
5^4 5 raised to the power of 4 5 raised to the fourth power
When a number is raised to the power of 2 or 3, 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 3^(12), the input should be 3 12.

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 1, 2, or 3 questions correctly.

Questions Answered Correctly Points Simplify
1 2 2
2 2* 2 4
3 2* 2 * 2 8
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 2^4 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 2 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 two is multiplied by itself three times.

2* 2 * 2_3 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. Base: & 2 Exponent: & 3 Now that the base and the exponent have been identified, the expression can be written. 2^3

b First, look at the given power expression for the bonus points earned if Tearrik answers four questions correctly.

2^4 Looking at the expression, it is possible to determine which is the base and which is the exponent of the expression. Base: & 2 Exponent: & 4 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. 2* 2 * 2* 2_4

c It was mentioned previously that if Tearrik answers three or four questions correctly, he would get 2^3 or 2^4 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.
2^5 In this expression, the number 2 is the number being multiplied and 5 is the number of times that 2 is multiplied by itself. Use this information to write the power expression as the product of repeated factors. 2^5 ⇕ 2* 2 * 2* 2 * 2_5 The resulting product of multiplying 2 by itself 5 times can now be determined. It might be helpful to start by calculating smaller products.
2* 2 * 2 * 2 * 2_5
AssociativePropMult

Associative Property of Multiplication

(2* 2) * (2* 2) * 2
Multiply

Multiply

4 * 4 * 2
AssociativePropMult

Associative Property of Multiplication

4 * (4 * 2)
Multiply

Multiply

4 * 8
Multiply

Multiply

32
Therefore, Tearrik gains 32 bonus points in the bonus question round if he answers five 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 2

The numbers that result from raising an integer to the power of 2 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
25 5 * 5 =5^2 Yes ✓ 5 is an integer.
30.25 5.5 * 5.5 = 5.5^2 No * 5.5 is not an integer.
32 5.656... * 5.656... = (5.656...)^2 No * 5.656... is not an integer.
64 8 * 8 =8^2 Yes ✓ 8 is an integer.
Discussion

Raising to the Power of 3

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
125 5 * 5 * 5=5^3 Yes ✓ 5 is an integer.
166.375 5.5 * 5.5 * 5.5 = 5.5^3 No * 5.5 is not an integer.
64 4 * 4 * 4=4^3 Yes ✓ 4 is an integer.
270 6.463... * 6.463... * 6.463... = (6.463...)^3 No * 6.463... 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 64 can be written as the square of 8.

8^2 = 64 Since 8 is an integer, 64 is a perfect square. To identify the numbers that are not perfect squares, consider the squares of 5 and 6. 5^2 = 25 6^2 = 36 Since 5 and 6 are integers, 25 and 36 are perfect squares. There are no integers between 5 and 6. This means that there are no perfect squares between 25 and 36.

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 27 is not a perfect square. Now, to determine if 136 is a perfect square, remember that the square of 10 is 100. The consecutive perfect squares are displayed in the table below.

Number Square Less Than, Greater Than, or Equal to 136?
10 10^2 = 100 100 < 136
11 11^2 = 121 121 < 136
12 12^2 = 144 144 > 136

As shown in the table, 136 is greater than 121 but less than 144. Since 136 lies between two consecutive perfect squares, 136 cannot be a perfect square. More perfect squares will be identified to determine if 225 is a perfect square.

Number Square Less Than, Greater Than, or Equal to 225?
13 13^2 = 169 169 < 225
14 14^2 = 196 196 < 225
15 15^2 = 225 225 = 225

The number 225 is the square of 15. Since 15 is an integer, 225 is a perfect square. The remaining number is 729. The square of 20 is 400. The square of 30 is 900. Since 729 is closer to 900 than to 400, it might be convenient to start with 30 and explore the next perfect squares in decreasing order.

Number Square Less Than, Greater Than, or Equal to 729?
30 30^2 = 900 900 > 729
29 29^2 = 841 841 > 729
28 28^2 = 784 784 > 729
27 27^2 = 729 729 = 729

Notice that 729 is a perfect square. Now that all numbers have been examined, the results can be summarized in another table.

Number Perfect Square?
64 Yes
27 No
136 No
225 Yes
729 Yes
b A perfect cube is a number that can be expressed as the cube of an integer. For example, the cube of 10 is 1000. Therefore, 1000 is a perfect cube.

10^3 = 1000 Since all the given numbers are less than 1000, any perfect cubes among them can be expressed as the cube of an integer from 2 to 9. Make a table to write these cubes and look for the given numbers, starting with 27.

Number Cube Less Than, Greater Than, or Equal to 27?
2 2^3 = 8 8 < 27
3 3^3 = 27 27 = 27

Because 27 can be expressed as the cube of the number 3, it is a perfect cube. Using a similar reasoning, the number 64 will be examined.

Number Cube Less Than, Greater Than, or Equal to 64?
4 4^3 = 64 64 = 64

As shown, 64 is a perfect cube. Now continue the table to see if 136 is a perfect cube.

Number Cube Less Than, Greater Than, or Equal to 136?
5 5^3 = 125 125 < 136
6 6^3 = 216 216 > 136

The number 136 lies between two consecutive perfect cubes, so 136 is not a perfect cube. Now the number 225 will be tested. It is a good thing that 216 is really close to 225!

Number Cube Less Than, Greater Than, or Equal to 225?
6 6^3 = 216 136 < 216
7 7^3 = 343 343 > 216

The number 225 lies between the perfect cubes 216 and 343, meaning that 225 is not a perfect cube. Finally, 729 will be tested.

Number Cube Less Than, Greater Than, or Equal to 729?
8 8^3 = 512 512 < 729
9 9^3 = 729 729 = 729

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

Number Perfect Cube?
64 Yes
27 Yes
136 No
225 No
729 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 -2 is squared, the result is a positive number! (-2) ^2 &= (-2)(-2) &⇕ (-2) ^2 &= 4 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, (-2)^6.

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 (-2) ^6 is multiplied by -2, the product is a negative number. (-2) ^6 * (-2) = 4 * 4 * 4 * (-2) Consider how exponents are written. Since -2 is being multiplied by itself 6+1=7 times, the expression can be written as (-2) ^7. This is an odd exponent. (-2) ^6 * (-2) = (-2) ^7 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


Level 1
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Powers
Exercises

Select the power that represents the given product.

7*7*7*7*7
11* 11*11 * 11 * 11 * 11 * 11

Looking at the product, we can see that 7 is being multiplied five times. 7* 7* 7* 7* 7_5 We need to determine which number is the base and which is the exponent to write the power expression. The base is the number being multiplied and the exponent is the number of times that the base is multiplied. Let's identify which is which! Base: & 7 Exponent: & 5 Now that we know which number is the base and which is the exponent, we can write the power that corresponds to this product of repeated factors. 7* 7* 7* 7* 7_5 ⇕ 7^5


Looking at the product, we can see that this time 11 is being multiplied seven times. 11* 11* 11 * 11 * 11 * 11 * 11_7 Similar to what we did before, we will identify the base and the exponent. Now that we know how to do it, let's write them! Base: & 11 Exponent: & 7 Now that we know which is the base and which is the exponent, we can write the power for this product. 11* 11* 11 * 11 * 11 * 11 * 11_7 ⇕ 11^7

E
Hint & Answer
S
Solution

Determine the value for each of the following powers.

7^4
13^2

We want to find the value corresponding to the given power. The first step that we should do is to look at the given expression. 7^4 In this expression, the number 7 is being multiplied by itself 4 times. Knowing this, we can write the power expression as a product of repeated factors. Then we can solve the multiplication expression. Let's do it!

Therefore, the expression 7^4 is equal to 2401.


Let's take another look at the given power expression. 13^2 We can answer this similarly to the way we did in Part A. First, we identify that 13 is being multiplied by itself 2 times. We can write a product of repeated factors to solve it. Let's do it!

Therefore, the expression 13^2 is equal to 169.

E
Hint & Answer
S
Solution
Which of the following numbers are perfect squares? Select all that apply.

A perfect square is a number that can be expressed as the square of an integer. For example, the number 64 can be written as the square of 8. 8^2 = 64 Let's test different numbers to determine which of the given numbers are perfect squares. To help us identify numbers that are not perfect squares, consider the squares of 5 and 6. 5^2 = 25 6^2 = 36 Since 5 and 6 are integers, 25 and 36 are perfect squares. Since there are no integers between 5 and 6, there are no perfect squares between 25 and 36.

We can use the squares of consecutive numbers to get consecutive perfect squares. If the number we are testing is between two consecutive perfect squares, it is not a perfect square. Let's look at the given numbers from least to greatest. 325, 961, 1158, 2809, 3249 Let's start our list of perfect squares with multiples of 10. Consider that the square of 10 is 100 and the square of 20 is 400. Since 325 is less than 400, we can look at the perfect squares below 400. Also, since 325 is closer to 400 than 100, we will focus the perfect squares of integers closer to 20 than 10.

Number Square Less Than, Greater Than, or Equal to 325?
20 20^2 = 400 400 > 325
19 19^2 = 361 361 > 325
18 18^2 = 324 324 < 325

This table shows that 325 is between two consecutive perfect squares, 324 and 361. This means that 325 is not a perfect square. Next, notice that the square of 30, 900, is close to 961. Let's find the squares of the numbers just after 30. Let's do it!

Number Square Less Than, Greater Than, or Equal to 961?
30 30^2 = 900 900 < 961
31 31^2 = 961 961 = 961

Since 961 is the square of 31, 961 is a perfect square! Next, let's see if 1158 is a perfect square. The square of 30 is 900 and the square of 40 is 1600. Since 1158 is a little closer to 900 than to 1600, we will look at the consecutive perfect squares after 30.

Number Square Less Than, Greater Than, or Equal to 1158?
30 30^2 = 900 900 < 1158
31 31^2 = 961 961 < 1158
32 32^2 = 1024 1024 < 1158
33 33^2 = 1089 1089 < 1158
34 34^2 = 1156 1156 < 1158
35 35^2 = 1225 1225 > 1158

Since 1158 lies between two consecutive perfect squares, 1158 is not a perfect square. Things are going along well. Now let's look at 2809. The square of 50 is 2500 and the square of 60 is 3600, so let's find the consecutive perfect squares greater than 2500.

Number Square Less Than, Greater Than, or Equal to 2809?
50 50^2 = 2500 2500 < 2809
51 51^2 = 2601 2601 < 2809
52 52^2 = 2704 2704 < 2809
53 53^2 = 2809 2809 = 2809

From the table, we can see that 2809 is the square of 53. Therefore, 2809 is a perfect square. Finally, let's examine 3249. It is a good thing that we already know that the square of 60 is 3600! Let's look at the squares of the integers less than 60.

Number Square Less Than, Greater Than, or Equal to 3249?
60 60^2 = 3600 3600 > 3249
59 59^2 = 3481 3481 > 3249
58 58^2 = 3364 3364 > 3249
57 57^2 = 3249 3249 = 3249

We found that 3249 is also a perfect square. Now we have examined all the given numbers. Good job! Let's summarize our results in a table.

Number Perfect Square?
961 Yes
325 No
2809 Yes
1158 No
3249 Yes
E
Hint & Answer
S
Solution
Which of the following numbers are perfect cubes? Select all that apply.

A perfect cube is a number that can be expressed as the cube of an integer. For example, the number 8 can be written as the cube of 2. 2^3 = 8 Let's test different numbers to determine which of the given numbers are perfect cubes. To help us identify numbers that are not perfect cubes, consider the cubes of 2 and 3. 2^3 = & 8 3^3 = & 27 Since 2 and 3 are integers, 8 and 27 are perfect cubes. Since there are no integers between 2 and 3, there are no perfect squares between 8 and 27.

We can use the cubes of consecutive numbers to get consecutive perfect cubes. If the number we are testing lies between two consecutive perfect cubes, then it is not a perfect cube. Let's look at the given numbers from least to greatest. 343, 526, 729, 2197, 2864 Let's start our list of perfect cubes with multiples of 10, which is 1000. The first three numbers are less than 1000 while the last two are greater. Let's look at the cubes of the integers less than 10 and compare them with the numbers in the list.

Number Cube Less Than, Greater Than, or Equal to 729?
10 10^3 = 1000 1000 > 729
9 9^3 = 729 729 = 729

From the table, we can see that the cube of 9 is 729, so 729 is a perfect cube. Things are going great! Let's continue this process to see if 526 is a perfect cube.

Number Cube Less Than, Greater Than, or Equal to 526?
9 9^3 = 729 729 > 526
8 8^3 = 512 512 < 526

In this case, we can see that 512 and 729 are consecutive perfect cubes. Since 526 lies between two consecutive perfect cubes, 526 is not a perfect cube. Let's continue until we can determine if 343 is a perfect cube.

Number Cube Less Than, Greater Than, or Equal to 343?
8 8^3 = 512 512 > 343
7 7^3 = 343 343 = 343

As we can see, 343 is the cube of 7. This means that 343 is a perfect cube. Now let's examine the numbers that are greater than 1000. The cube of 20 is 8000, which is much greater than 2197. Let's look at the perfect cubes closer to 1000 until we find 2197.

Number Cube Less Than, Greater Than, or Equal to 2197?
10 10^3 = 1000 1000 < 2197
11 11^3 = 1331 1331 < 2197
12 12^3 = 1728 1728 < 2197
13 13^3 = 2197 2197 = 2197

We found that 2197 is also a perfect cube! The only number left to consider is 2864. We will continue this process to figure out if this number is a perfect cube.

Number Cube Less Than, Greater Than, or Equal to 2864?
13 13^3 = 2197 2197 < 2864
14 14^3 = 2744 2744 < 2864
15 15^3 = 3375 3375 > 2864

Finally, we can see that 2864 is not a perfect cube. We finished examining the given numbers. Good job! Let's summarize our results in a table.

Number Perfect Cube?
2197 Yes
729 Yes
526 No
2864 No
343 Yes
E
Hint & Answer
S
Solution

The edge length of a cube is of 6 centimeters.

A cube with side lengths of 6 centimeters
The volume of the cube is given by the following power expression. 6^3 What is the value of the volume of the cube?

The volume of a cube is found by raising the edge length to the third power. This is why we are given this power expression for the volume of the cube with a side length of 6. 6^3 To find the value of this power expression, recall that a power is the product of repeated factors. The base of the power is the repeated factor. The exponent of the power indicates the number of times the base is used as a factor.

Since 6^3 is a power, we can rewrite it as the product of repeated factors. In our case, the base is 6 and the power is 3. This means that 6 will be used as a factor 3 times. Let's find the value of this power by writing the expression and then calculating the product.

Therefore, the volume of the cube is of 216 cubic centimeters.

E
Hint & Answer
S
Solution

Powers

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