Writing and Evaluating Expressions with Exponents 04:22 minutes
Transcript Writing and Evaluating Expressions with Exponents
Ever since she was a little girl, Leelee's wanted to be famous, but now that she’s a teenager, she realizes that she’s not especially good at anything. Poor Leelee, she realizes she may have an impossible dream, but one day, by complete accident, she made the most amazing video. Really, it's amazing! Leelee thinks that this video is her ticket to stardom and wonders, how long will it take until the video will go viral? Let’s help Leelee figure it out by using expressions with exponents. Leelee has a plan, as soon as her amazing video uploads, she'll send it to three friends and ask each friend to send it to three friends, and then ask each of those friends to send it to three of their friends, and so on, and so on. The video is really short, so Leelee figures that it should only take a minute for each of her three friends to watch the amazing video and then share it with three of their friends, and so on, and so on. Let’s write this information in a chart. After the first minute, her three friends will have watched the video, so that makes three views. After they each share it with three friends, that'll be another nine views. And, when those friends each share it with three more, that'll be another 27 views. Oh boy! By minute four, those 27 friends will each share with three of their friends, so that’s another 81 more views. Do you see a pattern? The number of views grows exponentially, so we can write each of these expressions using a base and an exponent. Let's take a look: 3 is equal to 31. 3 times 3 is the same as 3 squared, which we all know is 9, and 3 times 3 times 3 is equal to 3 cubed or 27, and so on, and so on. For each of these exponent expressions, the base, the number we multiply, remains the same. But the exponent, the number of times we multiply the base with itself, increases by one each time. What’s really neat is you can use this pattern to write an expression to calculate how many new views there will be any given minute. We can write this as 3 raised to the 'x' power, with ‘x’ representing the given minute. Leelee is impatient. She wants to be super famous ASAP. But what if she shares the video with five people, and they each share it with five people, and so on, and so on. At minute 10, how many new views will there be? If we multiply 5 by itself 10 times, that's the same as 5 raised to the 10th power! Calculating this out, there will be 9,765,625 people watching her video at minute 10. WOW! Leelee is really psyched! The video has finally finished uploading! She's gonna watch it so that she can be the very first viewer of this soontobefamous video. Soooo cool! Just like Leelee predicted, the video went viral, and the fly is really famous, but Leelee  not so much.
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Show how to write the given situation as an exonential expression.
Tipps
Each of the three friends recommends the video to another three friends.
Each indicates multiplication.
$a^n$ is a shorter way to write a multiplication: $a^n=a\times a\times ...\times a$, or $a$ multiplied by itself $n$ times.
If Leelee wants to know how many people are viewing her video in the fifth minute, she calculates $3^5=243$.
Lösung
She plans to send it to three friends, and after a minute she wants each of those friends to send it to another three friends each, and so on. Let's see what Leelee´s results should be:
 During the first minute, three people will view the video.
 During the second minute, $3\times 3$ people will view the video, which simplifies to $9$ people total.
 During the third minute, $3\times 3\times 3$ people will view the video, which simplifies to $27$ people total.
 During the fourth minute, $3\times 3\times 3\times 3$ people will view the video, which simplifies to $81$ people total.
 We can generalize this by saying that during the $x^{th}$ minute, $3^x$ people will view the video.

Label the base as well as the exponent.
Tipps
The exponent is the number of times you multiply the base by itself.
In general a power is given by $a^n$, where $a$ stands is the base of the power.
Keep the meaning of the corresponding positions in mind.
In the example beside $7$ is the base while $5$ is the exponent.
Lösung
In general a power is given by $a^n$, where $a$ is called the base and $n$ is called the exponent.
You can read it as $a$ raised to the power of $n$.

Express the following problem as a mathematical expression.
Tipps
We have that $5^3=5\cdot 5\cdot 5\cdot 5$, while $3^5=3\cdot 3\cdot 3\cdot 3\cdot 3$.
Five friends who should send it to five friends each gives us $5\times 5=5^2=25$ people viewing the video.
You finally have to multiply $25$ by $5$ once again: $25\times 5=125$.
Check this result with each of the given alternatives.
Lösung
First Leelee sends the videos to five friends. Then all of those friends send the video to five friends each. This leads to $5\times 5=5^2=25$.
Then, all of these $25$ people send the video to five friends each. So we finally get $5\times 5\times 5=5^3=125$ people viewing the video in the end.

Determine the corresponding expression.
Tipps
The volume of any rectangular prism is given by
width $\times$ length $\times $ height.
You can simplify as follows:
$(3x)\times(2x)\times(4x)=(3)(2)(4)(x^3)=24x^3$.
Just multiply the coefficients.
The volume of a cube with the length $4$ is given by $64$.
Lösung
The volume of a cube can be determined by raising the side to the power of $3$. This leads to $a^3$.
The volume of a rectangular prism is given by width $\times$ length $\times $ height. Thus, we get $(3h)(4h)(h)=(3)(4)h^3=12h^3$.
The area of a square is given by the length squared: $s^2$.

Explain how to transform the word problems.
Tipps
Here you see how a video is shared after $1$, $2$, $3$, and $4$ minutes, when it is shared with three people in the first minute, and those three people share it with three more people in the second minute, and so on.
Keep the notation for exponents in mind:
 $3=3^1$
 $3\times 3=3^2$
 $3\times 3\times 3=3^3$
 $3\times 3\times 3\times3=3^4$
So after $3$ minutes the video is shared with $3^3=27$ people.
Lösung
In each of the given examples you have to find the base as well as the exponent of a power:
Ladybugs
The video is shared with $4$ people. All of those share it with $4$ people each after one minute and so on. So after 5 minutes, we get $4\times 4\times 4...\times 4=4^5=1024$.
Butterflies
Leelee shares the video with $10$ people. All of those share it again with $10$ people each after one minute. And so on. Thus, after 4 minutes, we get $10\times 10\times 10...\times 10=10^4=10000$.
Grasshoppers
This time Leelee sends the video to $7$ people and asks them to send it out to $7$ people each after one minute. Then the video will be sent to $10$ people and so on. So after 6 minutes, we have $7\times 7\times 7...\times 7=7^6=117649$.

Find the right exponential expression.
Tipps
The base of $a^n$ is $a$.
The exponent of $a^n$ is $n$.
For example, $4\times 4\times 4=4^3$.
Lösung
New Years Messages
She sends the messages to $6$ friends. After one hour each of those friends sends a message to another $6$ friends. So it's $6\times 6$. After another hour all of those $6\times 6$ friends send a message to $6$ friends each ... So we get $6^4$ messages sent after three hours.
Bacterium
A bacterium doubles after one period to $2=2^1$ bacteria. After another period there are $2^2$ bacteria. So we can see that $2^n$ there are bacteria after $n$ periods.
Giraffe Legs
First determine the number of giraffes: $4$ giraffes in $4$ zoo exhibits leads to $4\times 4$ giraffes. Each of the giraffes has $4$ legs. So, in total we can count $4\times 4\times 4=4^3$ legs.
Mateo's Flyer
To promote his new taco bar, Mateo distributes flyers: first to $5$ people. All of those people distribute flyers to $5$ people each. After $4$ distributions $5^4$ flyers are given away.