anonymous
  • anonymous
HELP PLEASE. Evaluate the expression. (2m^2/n^3)-2 = n^6/4m^4 How did they get this??!?!
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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terenzreignz
  • terenzreignz
The one on the right is the answer, I suppose? :D
terenzreignz
  • terenzreignz
\[\Large \left(\frac{2m^2}{n^3}\right)^{-2}=\color{blue}{\frac{n^6}{4m^4}}\]
anonymous
  • anonymous
How did they get the answer though?

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anonymous
  • anonymous
Why can't it stay as 4m^4n^6? Why would the n^6 be at the the numerator and the 4m^4 be at the denominator?
terenzreignz
  • terenzreignz
Let's do it one step at a time ^_^ First, just like how multiplication distributes over addition and subtraction, EXPONENTS distribute over multiplication and division. So, we have to give each multiplied term in the fraction an exponent of -2. \[\Large \left(\frac{2m^2}{n^3}\right)^{-2}=\color{blue}{\frac{2^{-2}(m^2)^{-2}}{(n^3)^{-2}}}\] Catch me so far?
anonymous
  • anonymous
Yes
terenzreignz
  • terenzreignz
So... your question... you're basically asking why the numerator and denominator switched places, yes? ^_^
anonymous
  • anonymous
Yes D:
terenzreignz
  • terenzreignz
Simple... the exponent is negative... :D \[\Large \left(\frac{2m^2}{n^3}\right)^{\color{red}{-2}}=\color{blue}{\frac{n^6}{4m^4}}\] Were it POSITIVE 2 instead, we'd have (as you'd expect) \[\Large \left(\frac{2m^2}{n^3}\right)^{\color{green}{2}}=\color{blue}{\frac{4m^4}{n^6}}\]
anonymous
  • anonymous
Im not sure I get what youre saying... Im sorry :(
terenzreignz
  • terenzreignz
Do keep in mind that what THIS \[\Large \left(\frac{2m^2}{n^3}\right)^{\color{red}{-2}}\]actually means is \[\Huge \frac1{ \left(\frac{2m^2}{n^3}\right)^{\color{green}{2}}}\]
terenzreignz
  • terenzreignz
Right? Laws of exponents for ya ^_^
anonymous
  • anonymous
Yes I know that part
terenzreignz
  • terenzreignz
Okay, great :) We can now distribute the 2-exponent: \[\Huge \frac1{ \frac{4m^4}{n^6}}\]
terenzreignz
  • terenzreignz
And of course, this is just \[\Large \frac{n^6}{4m^4}\] Understood? ^_^
anonymous
  • anonymous
Okay, I get it so far
terenzreignz
  • terenzreignz
So far? It's already done ^^^
anonymous
  • anonymous
So it would just magically put n^6 to the top and rest at the bottom?
terenzreignz
  • terenzreignz
No, ... uhh Recall that \[\Huge \frac{\color{white}{ \ \ }1\color{white}{ \ \ }}{\frac{a}b}= \frac{b}a\]
anonymous
  • anonymous
What exponent law is this? Im so sorry! Is this the power of a quotient rule?
terenzreignz
  • terenzreignz
This isn't really a law of exponent, it's just evaluating complex fractions.
anonymous
  • anonymous
So once it turns out to 1/a/b the a/b would flip so it would be b/a ?
terenzreignz
  • terenzreignz
Put it this way... from elementary school division of fractions: \[\Huge \frac{ \ 1 \ }{\frac{a}b}= 1 \div \frac{a}b\] Now, to divide by a fraction, you multiply by its reciprocal: \[\Large = 1 \times \frac{b}a\] \[\Huge = \color{blue}{\frac{b}a}\]
terenzreignz
  • terenzreignz
Now, the same concept applies readily here: \[\Huge \frac{ \ \ 1 \ \ }{ \frac{4m^4}{n^6}}\]
anonymous
  • anonymous
Ohhhhh, okay!!! THANK YOU SO MUCH
terenzreignz
  • terenzreignz
Another way to look at this and perhaps a more useful way:
terenzreignz
  • terenzreignz
\[\Large \left(\frac{2m^2}{n^3}\right)^{-2}=\color{blue}{\frac{2^{-2}(m^2)^{-2}}{(n^3)^{-2}}}\]
terenzreignz
  • terenzreignz
So, laws of exponents dictate that this becomes \[\Large \frac{2^{-2}m^{-4}}{n^{-6}}\] right?
terenzreignz
  • terenzreignz
Well, you can think of negative exponents as a sign that the term 'doesn't belong there' Meaning, a negative exponent in the numerator means it should be in the denominator (with a positive exponent, of course) And vice versa. So, to bring each term 'where they belong', we get \[\Large = \frac{n^6}{2^2m^4}=\frac{n^6}{4m^4}\]
anonymous
  • anonymous
This one I got for sure, THANK YOU! :) :) :)
terenzreignz
  • terenzreignz
Good. Practice ^_^
terenzreignz
  • terenzreignz
A fair warning, however. You probably won't be dealing with this sort until later, but best be prepared... \[\Large \frac{a^3 + u^{-2}}{x^5}\] does \(\large \color{red}{not}\) equate to \[\Large \frac{a^3}{x^5\color{green}{+u^2}}\] You may ONLY move around the terms if they are multiplied, not if they are added or subtracted. Just a heads up ^_^
anonymous
  • anonymous
You just saved my life, thank you so much! I appreciate the help, sorry since I'm a slow learner :(
terenzreignz
  • terenzreignz
Faster than you think ^_^ Signing off now :D ------------------------- Terence out

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