Simplify the following expressions. Give your answers in power notation where the powers are positive.
8 to the power of -6 x 8 to the power of 2
6 to the power of 7 divided by 6 to the power of 12

- raphaelll

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- schrodinger

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- zepdrix

So the first one is this, ya? \(\large\rm 5^5\cdot5^5\)

- raphaelll

Yup thats the first one

- zepdrix

Here is our rule of exponents that will be very helpful to use:
\(\large\rm x^a\cdot x^b=x^{a+b}\)

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- zepdrix

Notice that when we apply this rule, the `base` stays the same.

- zepdrix

Any ideas how we can use this? :)

- raphaelll

umm, i just take 5 and add the two powers together?

- zepdrix

\[\large\rm 5^\color{orangered}{5}\cdot5^\color{orangered}{5}=5^{\color{orangered}{5+5}}\]Mmm good!

- raphaelll

thanks!
how do i go about doing the second question since there is a negative power?

- zepdrix

8 to the power of -6 x 8 to the power of 2
\[\large\rm 8^{-6}\cdot8^{2}=8^{-6+2}\]Same thing, addition :)
Remember how to add a negative and positive together?

- raphaelll

oh i see, is it also just addition if division is the middle step?

- zepdrix

division is a little different: \(\large\rm x^a\div x^b=x^{a-b}\)

- zepdrix

6 to the power of 7 divided by 6 to the power of 12
\[\large\rm 6^7\div6^{12}=6^{7-12}\]
Make sure you write the 7 and 12 in the correct locations.
I was forced to write the 7 first because that shows up on the left of the division.
The 7 is smaller, so yes, you should end up with a negative power after you complete the subtraction.

- raphaelll

okay thank you! so for multiplication it is just adding the two powers, and for division it is subtracting the two powers

- zepdrix

Yes!
You might be able to see a connection between the multiplication and division... if you really think about it. Not a big deal though :)

- raphaelll

you wouldn't happen to know anything about scientific notation would you? it's the last two questions on my assignment that i don't understand

- zepdrix

Can you write out the question? :o
That might make it easier for us.

- raphaelll

26.9 x 463 000
and
55.18 divided by 620

- zepdrix

For the first one, when you do the multiplication you get: \(\large\rm 12,454,700\)
This is not scientific notation.
For scientific notation, we want to write this as a number... times some power of 10.

- zepdrix

|dw:1441616656957:dw|So we want to write the number like this.
The number that goes here \(\large\rm \square\) should be a single digit. something larger than 1, and less than 10.
So when we look at our number, we can see that THIS is the number that goes in the square:

- zepdrix

|dw:1441616745335:dw|

- zepdrix

I'm going to write a decimal after that number,|dw:1441616795550:dw|

- zepdrix

The number that goes in the \(\large\rm \triangle\) is the number of places that you moved the decimal.

- zepdrix

|dw:1441616863094:dw|Decimal always starts here by the way.

- zepdrix

|dw:1441616886977:dw|Looks like we had to move the decimal 7 places.

- zepdrix

|dw:1441616919559:dw|Bam ok we got the first one done!

- zepdrix

Any confusion there? :o

- raphaelll

no actually you explained that really well!

- zepdrix

55.18 divided by 620
This one is a little different...
doing the division we end up with: \(\large\rm 0.089\)

- zepdrix

So what number goes in the \(\large\rm \square\), what do you think? :o

- raphaelll

8 because it is greater than 1 but less than 10

- zepdrix

Ok good! We want to keep all of the important values that follow the 8 as well.\[\LARGE\rm 8.9\times10^?\]

- raphaelll

10 to the power of 3?

- zepdrix

|dw:1441617458964:dw|Hmm, I don't think we moved 3 place values! :)

- raphaelll

oh yup my bad, i looked at the diagram from before thinking the decimal started from the end

- zepdrix

So, what's different about this problem is that ...
since we're moving the decimal to the `right`,
the exponent on the 10 will be `negative`.

- raphaelll

okay so if the decimal is moved to the right it is negative whereas to the left is positive, so the answer is 10 to the power of 2

- raphaelll

negative 2*

- zepdrix

\[\LARGE\rm 8.9\times10^{-2}\]Yay good job team \c:/

- raphaelll

thank you so much for all your help! will definitely be in touch if i ever need more help :P

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