Question

simplify 9^1/2 * 9^1/2

Answer 1

@PeterPan is it 81 because the fractions equal 1 so its just simple multiplying.

Answer 2

This is tricky...
But remember the laws of exponents... when you multiply two things with the same bases, just add their exponents...
\[\Large a^ma^n = a^{m+n}\]

So...
\[\huge 9^{\frac12}\cdot 9^{\frac12}=9^{\frac12+\frac12}\]
And we know two halves make a whole, so...\[\huge 9^{\frac12}\cdot 9^{\frac12}=9^{\frac12+\frac12}=9^1\]

Answer 3

Thats crazy talk peterpan. but i trust you.

Answer 4

so its 9

Answer 5

dont leave meee

Answer 6

im stranded.

Answer 7

Sorry...
yes, it's 9 :)

Answer 8

okay thank you. i just didnt want to be wrong. and fail.

Answer 9

I mean, when you really think about it,
\[\Large 9^{\frac12}= \sqrt 9 = 3\]
Effectively...
\[\Large 9^{\frac12}\cdot 9^{\frac12}=3\cdot 3= 9\]

Answer 10

By the way, that MIGHT come in handy for you, to know by instinct that square root is exactly the same as raising to an exponent 1/2

Answer 11

so if its 3 the its 3.5

Answer 12

And equivalently, cube root is raising to an exponent 1/3, fourth root is raising to an exponent 1/4.... etc

Answer 13

I'm sorry, what?

Answer 14

Where did the 5 come from?

Answer 15

three and a half.

Answer 16

yeah, why? Why 3 and a half?

Answer 17

What do you mean by "if its 3..." ?

Answer 18

the exponient is 3

Answer 19

Oh, there is no 3 exponent...
\[\Large base^{exponent}\]

Answer 20

so raising it a half would be 3.5 correct?

Answer 21

logically.

Answer 22

You mean...
\[\Large 3^{\frac12 }=3.5\]?

Answer 23

yes.

Answer 24

Then no. :P
\[\Large 3+\frac12 = 3.5\qquad\color{green}{\checkmark}\]
\[\Large 3^{\frac12 }= 3.5 \qquad \color{red}{\bf{\times}}\]

Answer 25

\[\Large 3^{\frac12}= \sqrt3 \approx 1.732\]

Answer 26

the demnoiter 2 so it would be 3*3

Answer 27

No...
\[\Large 3\cdot 3 = 3^2 \ne 3^{\frac12}\]

Answer 28

In other words, 3*3 is 3^2
and NOT 3^(1/2)

Answer 29

so if its 3 1/2 i divide 3 by 2

Answer 30

Multiplying the exponent to the base?
\[\Large 3^{\frac 12}= 3\times \frac12 = \frac32 \qquad \color{green}{?}\]

Answer 31

My teacher calls that step the "never ever do that or I'll kill you" step...
you don't multiply the exponent to the base...

Answer 32

Just take it from me, the exponent 1/2 means you take the square root of the base...
\[\huge a^{\frac12}= \sqrt a\]

Answer 33

and nothing else...

Answer 34

so the 1 just vanishes?

Answer 35

Well, technically
\[\huge a^{\frac12}= \sqrt{a^1}\]
but, clearly, a^1 is just a.

Answer 36

okay what if (8x^-6) ^1/3

Answer 37

Okay, two new rules come into play, first, when you raise the product of two things to an exponent, you just distribute the exponent over the product...
\[\Large (ab)^m = a^mb^m\]

Answer 38

So, with that in mind, we get
\[\huge (8x^{-6})^{\frac13 }= 8^{\frac13}\cdot (x^{-6})^{\frac13}\] yeah?

Answer 39

so 8 cubed times -6x cubed

Answer 40

nu-uh.... does that look like a cube to you?
THIS is what a cube looks like :D

no, seriously, a cube is an exponent of 3, NOT 1/3

an exponent of 1/3 means cube ROOT.

Answer 41

So, what is the cube root of 8? :)

Answer 42

2.66666 or 2.7

Answer 43

<whistles>
that's the square root (read in an annoying childish sing-song voice)

Answer 44

face palm

Answer 45

So... what IS the cube root of 8, then? :)

Answer 46

8 divided by 3 right???

Answer 47

you're doing it again... multiplying the exponent to the base.... my teacher's gonna kill you :D

Answer 48

shh,. dont tell on me

Answer 49

lol, in all seriousness, the exponent isn't something you multiply (or divide) with the base... they're off limits from each other, if you know what I mean...

Answer 50

The cube root of 8, is that number, which, when you multiply to itself three times, you get 8.
r x r x r = 8

Answer 51

512

Answer 52

dont tell me im wrong again...

Answer 53

Well, you're not right :P
You really need to see that distinction between cube and cube ROOT. LOL
The cube of 8 is 512, because 8 x 8 x 8 = 512...
however, the cube ROOT of 8 is that number such that when you multiply to itself three times, you get 8.
Take for example, 27, like earlier... the cube ROOT of 27 is 3, because 3x3x3 = 27

^.^

Answer 54

so its 24.

Answer 55

And you tried dividing 8 by the exponent, 1/3... -.-

Answer 56

No... suffice it to say, the cube root of 8 is 2... because


\[\Large 2^3 = 2\times 2\times2 = 8\]

Answer 57

no. draw it out for me peterpan.

Answer 58

So, everything understood, the cube root of 8 is 2?

Answer 59

2.66666

Answer 60

You multiplied 8 by 1/3 again?

Answer 61

Pull your act together :D
If in doubt, check if 2.66666 is indeed the cube root of 8, by *cubing* it...
2.66666^3 is like, 18.96.... definitely not 8 :P

Answer 62

OOOOOOOOHHHHHHH!!!!!! that waht i wasnt putting together.

Answer 63

so... the cube root of 8?

Answer 64

eight divided by 3

Answer 65

^3

Answer 66

You multiplied 8 by 1/3 again...
three strikes, you're out :D

Answer 67

i got 72

Answer 68

no i divided.

Answer 69

i got 2.66666

Answer 70

dividing by 3 is the same as muliplying by 1/3.
and either way, it's wrong...

come on, Chris, how many times do I have to tell you that you don't multiply or divide the base with the exponent?

Answer 71

:D

Answer 72

\[\Large 8^{\frac13}\ne 8\div 3\]\[\large \color{red}{!!!!!!!!!!!}\]

Answer 73

so its 8x8x8?

Answer 74

<awkward smile>
no....
\[\Large 8\times 8\times 8 = 8^3 \ \ne \ 8^{\frac13}\]

Answer 75

so what is 8 1/3

Answer 76

I actually already dropped it :)
\[\Large 8^{\frac13}= 2\]
because....
\[\huge 2\times2\times2 = 8\]

Answer 77

so its backwards sorta say.

Answer 78

so its 2/x^2

Answer 79

hmmmn....I realise that doing things backwards is harder than forwards... (like how I struggled with subtraction after mastering addition ^.^)

But it's something you need to learn :P

And unfortunately, this is only part of it...
\[\huge (8x^{-6})^{\frac13 }= \color{red}{8^{\frac13}}\cdot (x^{-6})^{\frac13}=\color{red}2\cdot (x^{-6})^{\frac13}\]

Answer 80

Now we need to deal with this bit...
\[\huge 2\cdot\color{blue} {(x^{-6})^{\frac13}}\]

Answer 81

Which brings us to another law of exponents...
\[\huge (a^m)^n= a^{mn}\]
When you raise a power to another power, you multiply the exponents...
So how do you handle this
\[\huge 2\cdot\color{blue} {(x^{-6})^{\frac13}}\]

Answer 82

i would multiply -6 * 1/3

Answer 83

That's right...
notice the difference... you may multiply exponents with other exponents, but NEVER with the base...

Answer 84

so -6/3?

Answer 85

mhmm... simplify?

Answer 86

-2?

Answer 87

BINGO .
\[\huge 2\color{blue}{x^{-2}}\]

Answer 88

So, is that in your choices?