Rational Exponents & Finding The Vol. Of A Prism.

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Hey there;

Entirely new here, needing some help understanding a few things... If anyone out there can be of help that'd be wonderful.

I'm doing this by correspondence and I came across something in the notes & lession portion concerning Rational Exponents and I'm sort of confused.

1) The first two questions I understood, though I'm not entirely sure why the question (x^2)^-3/2 is saying that to work out the answer you have to have (sq. root x^2)-3 whereas the above question which was (x^8)^-1/2 was worked out like this:

x^8^(-1/2)
x^-8/2
x^-4
1/x^4 ---> All of that I get...

Though, I'm not sure why the other question (x^2)^-3/2 is worked out as followed:

(sq. root x2)^-3
x^-3
=1/x^3

(Not sure how or even why they put the sq. root in there?)

2) My second dilemnia concerns trying to find the volume of a prism by using the area of a triangle and the length of the prism which is something like this:

(1/2[4sq.root 3][2 sq.root 5])(3 sq. root 15)
(4/2 sq. root 3 x 2/2 sq. root 5) (3. sq. root 15)
etc., etc.,
=180 m^3

This is the part where I am stuck, it says that the its. (2x2 sq. root 15) in the first brackets; though if 2/2 is 1.. where are they getting the 2nd 2 from?? Don't you multiple the 1/2 to the two remaining brackets within the 1st bracket?

If I'm completely brainless then my sincerest apologizes, I'm just trying to make some sense out of it. So, if anyone could help me that'd be great.
 
Hello, Angelfirebaby71!

If it's not too late, I'd get another correspondence course.
. . This one is doing strange things . . .

(x<sup>8</sup>)<sup>-1/2</sup> .= .x<sup>-8/2</sup> .= .x<sup>-4</sup> .= .1/x<sup>4</sup> . . . All of that I get

I'm not sure why the other question (x^2)^-3/2 is worked out as follows:
. . . . . . . . . . ._
(x<sup>2</sup>)<sup>-3/2</sup> .= .(√x<sup>2</sup>)<sup>-3</sup> .= .x<sup>-3</sup> .= .1/x<sup>3</sup>

(Not sure how or even why they put the sq. root in there)
Neither am I . . . what a screwy thing to do!

Why didn't they do the first one the same way?
. . . . . . . . . . . . ._
. . (x<sup>8</sup>)<sup>-1/2</sup> .= .(√x)<sup>-8</sup> .= .x<sup>-4</sup> .= .1/x<sup>4</sup> . . . and confuse everyone?


2) My second dilemnia concerns trying to find the volume of a prism
by using the area of a triangle and the length of the prism
area of what triangle ... and what is the length of a prism?
which is something like this:

(1/2[4sq.root 3][2 sq.root 5])(3 sq. root 15)
(4/2 sq. root 3 x 2/2 sq. root 5) (3. sq. root 15)
etc., etc., . . . . . really stupid!
=180 m^3
I'm not sure where they learned their arithmetic, certainly not on this planet.
. . . . . . . . . . . . . . . _ . . . _ . . .__
We have: .(1/2)(4√3)(2√5)(3√15)

Get the rational numbers together: .(1/2)(4)(2)(3) .= .12
. . . . . . . . . . . . . . . . . . . . . . . ._ . ._ . .__ . . . . ___
Get the radicals together: . (√3)(√5)(√15) .= .√225 .= .15

And we get: .12 x 15 .= .180
 
Errr...

1) So, no hope in understanding why the sq. root then? As that way
. (x8)-1/2 .= .(√x)-8 .= .x-4 .= .1/x4 confuses me...


2) the area on the diagram of the triangle is 2√5 & 4√3 The length of the prism is the (3√15) and the answer which I didn't bother to write out fully (my mistake, sorry :()

[1/2(4√3)(2√5)][3√15]
[(4/2√3)(2/2√5)][3√15]
[(2√3)(2√5)][(3√15)] <- where are they getting the 2√5 from tho?
[4√15][3√15]
12 x 15
180 m^3
 
Boy, Firebaby, you're sure not being very clear...

Anyway:

Though, I'm not sure why the other question (x^2)^-3/2 is worked out as followed:
(sq. root x^2)^-3
x^-3
=1/x^3
(Not sure how or even why they put the sq. root in there?)

What "they" did was change the -3/2 to 1/2 * -3;
(x^2)^(1/2) is same as sqrt(x^2); and sqrt(x^2) = x; ok?

"They" could have done it this way:
(x^2)^-3/2 : that needs another set of brackets:
(x^2)^(-3/2) : do you see why?
= x^(2 * (-3/2))
= x^(-3)
= 1 / x^3

That bunch of "they" should be shot at sunrise for doing it "their" way!

...................................................................................................
(1/2[4sq.root 3][2 sq.root 5])(3 sq. root 15)
(4/2 sq. root 3 x 2/2 sq. root 5) (3. sq. root 15) etc., etc.,
=180 m^3
This is the part where I am stuck, it says that the its. (2x2 sq. root 15) in the first brackets; though if 2/2 is 1.. where are they getting the 2nd 2 from?? Don't you multiple the 1/2 to the two remaining brackets within the 1st bracket?

Boy, no fun trying to explain that by typing...well:
(1/2[4sq.root 3][2 sq.root 5])(3 sq. root 15) : show 4sq.root 3 as 4sqrt(3)
Above means:
[4sqrt(3) times 2sqrt(5)] divided by 2, then the result of that is multiplied by 3sqrt(15); so:

{[4sqrt(3)][2sqrt(5)] / 2}3sqrt(15)
= [2sqrt(3)][sqrt(5)][3sqrt(15)]

So 180m^3 = [2sqrt(3)][sqrt(5)][3sqrt(15)]
m^3 = [2sqrt(3)][sqrt(5)][3sqrt(15)] / 180

m = {[2sqrt(3)][sqrt(5)][3sqrt(15)] / 180}^(1/3)
m = (90 / 180)^(1/3) : YES, that does work out to 90!
m = (1/2)^(1/3)
m = .7937~

...what can I say :(

Well damn it all: didn't see Soroban's and your last post;
hope I've helped anyway!
 
Uhhh

Now, I'm officially lost. :(

1) at top it says to Simplify each expression and assume all variables are positive. If that helps any.... :-S
 
Why not one more???
You can and should do the
(x²)^(-3/2) the same as the
(x^8)^(-1/2), which you say you understood.
You multiply the 2 by the -3/2 to get -3 for
x^(-3) which =
1/x³
As long as you understand that sqrt(x²) and (x²)^(1/2) are the same thing, don't worry about why they chose to make the unnecessary substitution.

The second is a triangular prism whose volume is area of the triangle times Length.
B=4*sqrt(3)
H=2*sqrt(5)
L=3*sqrt(15)
The area of the triangle is (1/2)Base*Height.
They had a typo in 2/2*sqrt(5). The stray /2 vanished from the third equation and didn't belong in the second.
(1/2)BH =
(1/2)*(4sqrt(3))*(2sqrt(5)) =
4sqrt(15)
OK so far?
Then we multiply that times L
(4sqrt(15))*(3sqrt(15)) =
12*15 =
180 cubic meters.
 
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