Question

HELP!!?? I will give a medal!
Part A: Use the properties of exponents to explain why 81/3 is called the cube root of 8. (5 points)

Part B: The length of a rectangle is 3 units and its width is √3 unit. Is the area of the rectangle rational or irrational? Justify your answer. (5 points)

Answer 1

@SolomonZelman

Answer 2

@perl

Answer 3

@DanJS

Answer 4

Someone help please

Answer 5

I doubt that 81/3 is a cube root of 8.

Answer 6

Well I don't know.. that's what the problem says.

Answer 7

I don't consider this question to be correct and complete (hope you know why).

Answer 8

8 and 1/3

Answer 9

\(\large\color{black}{ 8\frac{\LARGE 1}{\LARGE 3} }\) is still not a cube root of 8....

Answer 10

or do you mean, \(\large\color{black}{ 8^{\frac{\LARGE 1}{\LARGE 3}} }\) ?

Answer 11

Okay well that's what it says. I didn't make up the problem my teacher did.

Answer 12

Yes

Answer 13

\(\large\color{black}{ 8^{1/3} }\), is better;)

Answer 14

Okay

Answer 15

Well, there is a rule: \(\Large\color{black}{ \color{red}{A}^{\color{green}{B}/\color{blue}{C}} =\sqrt[\color{blue}{C}]{\color{red}{A}^\color{green}{B}} }\)

Answer 16

Apply this rule, to \(\Large\color{black}{ \color{red}{8}^{\color{green}{1}/\color{blue}{3}} }\)

Answer 17

go ahead..

Answer 18

don't be afraid to interrupt me.
If you have problems with this, ask.

Answer 19

i can't type it out like that on my computer I have to type it out..
3 square root 8 exponent 1...
it probably doesn't make sense but I don't know how to word it

Answer 20

I can't type it out like you can

Answer 21

you mean, \(\Large\color{black}{\sqrt[\color{blue}{3}]{\color{red}{8}^\color{green}{1}} }\) ?

Answer 22

yes

Answer 23

(it is just codes, if you want we can spend some time learning THAT as well.
Just if you wish)

Answer 24

Anyways, now we will finish the problem.

Answer 25

Okay

Answer 26

Oh, what a pain, excuse me, I lost connection for a second.

Answer 27

When you say, \(\Large\color{black}{ \sqrt[\color{blue}{3}]{\color{red}{8}^\color{green}{1}} }\) what number does absolutely not matter here?

Answer 28

1

Answer 29

yes, so \(\Large\color{black}{ \sqrt[\color{blue}{3}]{\color{red}{8}^\color{green}{1}} }\) is same as, \(\Large\color{black}{ \sqrt[\color{blue}{3}]{\color{red}{8}} }\)

Answer 30

Right

Answer 31

@SolomonZelman

Answer 32

So all we did is: \(\Large\color{black}{ \color{red}{8}^{\color{green}{1}/\color{blue}{3}}=\sqrt[\color{blue}{3}]{\color{red}{8}^\color{green}{1}}=\underline{\sqrt[\color{blue}{3}]{\color{red}{8}}} }\)

Answer 33

I am here. It is just a glitch that isn't showing that.

Answer 34

See the underlined part, what would you call it?

Answer 35

Are we dividing that by something?

Answer 36

just tell me what the underline part is called please.

Answer 37

I don't know

Answer 38

" cube root of eight " , ain't it so ?

Answer 39

\(\Large\color{black}{ \sqrt[\color{blue}{3}]{\color{red}{8}} }\) - CUBE ROOT OF EIGHT.

Answer 40

Okay...

Answer 41

God punshes me with the connection, sorry again

Answer 42

That's fine.

Answer 43

Grammar... anyways,

Basically we did:

\(\Large\color{black}{ \color{red}{8}^{\color{green}{1}/\color{blue}{3}}=\sqrt[\color{blue}{3}]{\color{red}{8}^\color{green}{1}}=\sqrt[\color{blue}{3}]{\color{red}{8}} }\)

Answer 44

So you can see that:

\(\Large\color{black}{ \color{red}{8}^{\color{green}{1}/\color{blue}{3}}}\) is the same thing as \(\Large\color{black}{ \sqrt[\color{blue}{3}]{\color{red}{8}} }\)

Answer 45

Right.

Answer 46

I think you are done:)

Answer 47

Any questions about part A?

Answer 48

No I don't think so :)

Answer 49

no questions, so we can move on to PART B ?

Answer 50

Yes

Answer 51

Okay, firstly, is \(\large\color{slate}{ \sqrt{3} }\) a rational or irrational number ?

Answer 52

(If you happened not to know a definition of any terms I am using, then please don't hesitate to ask)

Answer 53

Irrational i am pretty sure..

Answer 54

@SolomonZelman

Answer 55

yes, correct.

Answer 56

Okay

Answer 57

length of a rectangle, lets see if I can draw it on here.... give me a sec.

Answer 58

\(\normalsize\color{royalblue}{ \rm FORMULA. }\)


\(\huge\color{magenta}{ \rm L }\)
\(\normalsize\color{ slate }{\Huge{\bbox[5pt, lightcyan ,border:2px solid black ]{ \color{lightcyan}{\Huge\frac{~~~~~~~~\frac{~}{~~~\frac{~\frac{}{}~}{~\frac{~}{~}}~~~~}~~~}{~\frac{\frac{~}{~\frac{~\frac{}{~}}{~}}~~}{~}~~} } }}}\) \(\huge\color{darkviolet}{ \rm W }\)


\(\huge\color{blue}{ \rm A_{rectangle} = \color{magenta}{ \rm L }\times\color{darkviolet}{ \rm W } }\)




\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)



\(\normalsize\color{royalblue}{ \rm IN~~YOUR~~CASE }\)


\(\huge\color{magenta}{ \rm 3 }\)
\(\normalsize\color{ slate }{\Huge{\bbox[5pt, lightcyan ,border:2px solid black ]{ \color{lightcyan}{\Huge\frac{~~~~~~~~\frac{~}{~~~\frac{~\frac{}{}~}{~\frac{~}{~}}~~~~}~~~}{~\frac{\frac{~}{~\frac{~\frac{}{~}}{~}}~~}{~}~~} } }}}\) \(\huge\color{darkviolet}{ \rm \sqrt{3} }\)


\(\huge\color{blue}{ \rm A_{rectangle} = \color{magenta}{ \rm 3 }\times\color{darkviolet}{ \rm \sqrt{3} } }\)

Answer 59

So, do you think your area is a rational number, or not?

Answer 60

I don't think so..

Answer 61

So the area is rational, or the area is irrational?

Answer 62

Irrational..

Answer 63

(Just want to verify what you think)

Answer 64

Okay

Answer 65

Yes, irrational. Correct!

Answer 66

Have any questions ?

Answer 67

Okay :) how am I suppose to justify it? I just am not sure how to explain it

Answer 68

interesting, I know it, but I came into a somewhat thick wall trying to actually lay it out in simple words, so that it is easy to understand, let me think how to explain it....

Answer 69

You can take it as a definition, that

\(\large\color{slate}{ \rm irrational~~number }\) \(\large\color{blue}{ \times }\) \(\large\color{slate}{ \rm (none-zero)~~rational~~number }\)

is an irrational number.

Answer 70

I think I figured it out :)

Answer 71

When you for instance have:
\(\large\color{slate}{ \sqrt{4}\times a}\) in order for the resulting product to be a rational number, then \(\large\color{slate}{ a}\) has to be \(\large\color{slate}{ \sqrt{4}}\) (this way you get 4), or \(\large\color{slate}{ a}\) has to be zero (this way you get 0).

Answer 72

Do you mind helping me with one more problem?

Answer 73

yes, maybe. Will see how my time goes.

Answer 74

It was nice helping you... btw, yw

Answer 75

I was going to say thank you....