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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

When a specific number is added to itself multiple times, the result is the same as multiplication.
*multiplied* by itself a certain number of times?

But what if, instead, the number is

Is there a short way to write this kind of multiplication expression? This will be explored in this collection.

Discussion

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 $4.$

In this example, $7$ is multiplied by itself $4$ times.$7_{4}=47⋅7⋅7⋅7 $

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 |

squaredor

cubed,respectively. Expressions for greater powers are all usually read as the last example on the table.

Pop Quiz

Example

Tearrik is participating on a trivia contest at school.

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.

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{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

c Tearrik manages to answer five questions correctly before the bonus round ends. How many bonus points does he win?

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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.

a Looking at the table, answering three questions correctly means the number $two$ is multiplied by itself $three$ times.

$32⋅2⋅2 $

To write this product as power, it is important to identify the $Base:Exponent: 23 $

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:Exponent: 24 $

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.
$42⋅2⋅2⋅2 $

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}⇕52⋅2⋅2⋅2⋅2 $

The resulting product of multiplying $2$ by itself $5$ times can now be determined. It might be helpful to start by calculating smaller products.
$52⋅2⋅2⋅2⋅2 $

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

$32$

Pop Quiz

Find the value of the given power.

Discussion

The numbers that result from raising an integer to the power of $2$ appear frequently in math. These numbers are called perfect squares.

Concept

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

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

Concept

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

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!

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.

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

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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.

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}=256_{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 $36.$
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

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

Closure

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, $(-2)_{6}.$
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 |