How did they do this simplification?

[benj1]

Hi everyone

having a bit of trouble going through a worked example, specifically in a section where they simplify the following

1.5L^-.7K^0.7 / 0.7L^0.3K^-0.3 = 1

therefore

1.5k^0.7-(-0.3) / 0.7L^0.3-(-0.7) = 1

part in bold is one I am having trouble with.

Any help appreciated

tried using LATEX but didn't work out at all :\
Thanks

 

[VietDao29]

Do you mean to simplify:
$$\frac{1.5 L ^ {-0.7} K ^ {0.6}}{0.7 L ^ {0.3} K ^ {-0.3}}$$?
So, do you know that:
$$\frac{\alpha ^ \beta}{\alpha ^ \gamma} = \alpha ^ {\beta - \gamma}$$?
And:
$$\alpha ^ {- \beta} = \alpha ^ {0 - \beta} = \frac{a ^ 0}{a ^ \beta} = \frac{1}{\alpha ^ \beta}$$
So applying that to the expression gives:
$$\frac{1.5}{0.7} \frac{L ^ {-0.7}}{L ^ {0.3}} \frac{K ^ {0.6}}{K ^ {-0.3}} = \frac{1.5}{0.7} L ^ {-0.7 - 0.3} K ^ {0.6 - (-0.3)} = \frac{1.5}{0.7} \frac{K ^ {0.9}}{L ^ {1.0}}$$.
Do you get it now?

 

[vsage]

Looks like they simply grouped similar powers in the numerator and denominator. When you move an exponent to the denominator you subtract that exponents power from the power of the same base that already exists in the denominator. It will be easier to see if you break up the division into something like $$\frac{1.5L^{-0.7}}{0.7L^{0.3}}\times\frac{K^{0.7}}{K^{-0.3}}$$.

 

[benj1]

thanks heaps for the responses viet and vsage.. much appreciated

viet: the top part you have k^0.6 - it should be k^0.7

and then the indices should cancel each other out

leaving you with

1.5K / 0.7L

Im completely lost with that tex stuff

 

[VietDao29]

This page contains 3 pdf pages that can help you understand and use LaTeX, it's in the General Physics board.
Just remember that you do not need '\' for numbers, or words, it's just used for functions.
Click on every LaTex image to see its code. For example, click on this one:
$$\frac{1.5 K}{0.7 L}$$

 

[benj1]

thanks again..
heres another problem

240=12L^0.5(L/2)^0.5

240=12/sqrt of 2 * L

any help.. again would be appreciated

 

[hypermonkey2]

thanks again..
heres another problem
$$240=12L^0.5\frac{\L}/{2}^0.5 240 =\frac{12}{\sqrt{2} * L$$
any help.. again would be appreciated

$$12L^{0.5} * \frac{L^{0.5}}{2^{0.5}}$$ is an equivalent expression to your first one since

this is just properties of exponents.
therefore you can reduce your first expression to
$$12 * \sqrt{L} * \sqrt{L} * \frac{1}{\sqrt{2}}=\frac{12}{\sqrt{2}} * L$$

hope this helps!

 

[benj1]

thank you hypermonkey!

i think my lack of sleep is getting the better of me or just my complete inability at maths.

still not really understanding the 12/sqrt of 2 part

sqrt to rid the equation of indices?

why has the 12 gone to the numerator of the fraction?

this is very basic and I am finding it too difficult... *sigh*

 

[hypermonkey2]

thank you hypermonkey!
i think my lack of sleep is getting the better of me or just my complete inability at maths.
still not really understanding the 12/sqrt of 2 part
sqrt to rid the equation of indices?
why has the 12 gone to the numerator of the fraction?
this is very basic and I am finding it too difficult... *sigh*

dont worry about it, we all start somewhere. just as long as you don't get discouraged, youll be mathematizing in no time.
heres what you might now realize
$$12 * \frac{1}{\sqrt{2}}=\frac{12}{1} *\frac{1}{\sqrt{2}}=\frac{12}{\sqrt{2}}$$
that is the law of fraction multiplication.
it makes sense too, since
$$2 * \frac{1}{2}=2 * 0.5=1=\frac{2}{2}$$
do you agree?

 

[benj1]

help here has been overwhelming thanks.. I think I am getting it, wheter its sinking in or not.. :(

the next step is proving a bit of a problem as well

240 = (12 / sqrt of 2) * L
L = sqrt of 2 * 240 / 12

again any help.. appreciated