| Author |
Message |
|
rockrock
|
Post subject: Simplifying exponents  Posted: Tue Jun 29, 2010 5:32 pm |
|
 |
| Course Students |
|
|
Posts: 38
|
|
One of my CAT exam problem simplifies:
9x^4 – 4y^4 = (3x^2)^2 – (2y^2)^2.
Can someone explain this rule for me? I simplified 9x^4 to be (3^2)x^4...and that got me nowwhere. But I'm trying to memorize the correct rule.
|
|
  |
|
 |
|
julien.pitteloud
|
Post subject: Re: Simplifying exponents Posted: Wed Jun 30, 2010 3:05 am |
|
|
| Forum Guests |
|
|
Posts: 5
|
|
Hello, nice to meet you.
also there is : (x^2)^2=x^(2*2)=x^4
c ya.
|
|
| |
|
|
|
adiagr
|
Post subject: Re: Simplifying exponents Posted: Wed Jun 30, 2010 3:59 am |
|
|
| Students |
|
|
Posts: 89
|
rockrock wrote: One of my CAT exam problem simplifies:
9x^4 – 4y^4 = (3x^2)^2 – (2y^2)^2.
Can someone explain this rule for me? I simplified 9x^4 to be (3^2)x^4...and that got me nowwhere. But I'm trying to memorize the correct rule. The rule used here is: m^2 - n^2 = (m-n) x (m+n)
Given expression has been written in the form of perfect squares so, 9x^4 = Square of (3x^2) [ Say m = 3.x^2] 4y^4 = Square of (2y^2) [ Say n = 2.y^2] Then we can apply above formula and rewrite the expression as: (3.x^2 - 2.y^2) x (3.x^2 + 2.y^2) Aditya
|
|
| |
|
|
|
julien.pitteloud
|
Post subject: Re: Simplifying exponents Posted: Wed Jun 30, 2010 4:36 pm |
|
|
| Forum Guests |
|
|
Posts: 5
|
|
Hello Aditya,
could we continue the rule for
(3x^2-2y^2)=(sqrt(3)x-sqrt(2)y)(sqrt(3)x+sqrt(2)y)
but do you know if this is sometimes used to iterate further (exponents 1/2, 1/4, aso) ?? This would have no sense.
|
|
| |
|
|
|
adiagr
|
Post subject: Re: Simplifying exponents Posted: Thu Jul 01, 2010 1:41 am |
|
|
| Students |
|
|
Posts: 89
|
julien.pitteloud wrote: Hello Aditya,
could we continue the rule for
(3x^2-2y^2)=(sqrt(3)x-sqrt(2)y)(sqrt(3)x+sqrt(2)y)
but do you know if this is sometimes used to iterate further (exponents 1/2, 1/4, aso) ?? This would have no sense. See, it will depend on the question. If you can post the entire question it will be easier to tell. Aditya
|
|
| |
|
|
|
tim
|
Post subject: Re: Simplifying exponents Posted: Thu Jul 08, 2010 2:07 pm |
|
|
| ManhattanGMAT Staff |
|
|
Posts: 1779
Location: Southwest Airlines, seat 21C
|
|
Yeah, you can do things like that, but it will VERY rarely be useful to take things to that extreme to solve a problem of this type..
_________________
Tim Sanders
Manhattan GMAT Instructor
|
|
| |
|
|
|
rockrock
|
Post subject: Re: Simplifying exponents Posted: Mon Aug 02, 2010 3:04 pm |
|
|
| Course Students |
|
|
Posts: 38
|
|
I still get a bit confused because when I use an example of same base same exponent it doesn't add up.
If 9x^4 ---> 3x^2 * 3x^2 (multiply bases and add the exponents)
then
same base,same exponent should also mean:
2^2 * 2^2 = 4^4.... (multiply the bases and add the exponents)
but instead the answer is 4^2 (retain the exponent, multiply the bases).
Where am I going wrong here?
|
|
| |
|
|
|
adiagr
|
Post subject: Re: Simplifying exponents Posted: Tue Aug 03, 2010 8:13 am |
|
|
| Students |
|
|
Posts: 89
|
rockrock wrote: If 9x^4 ---> 3x^2 * 3x^2 (multiply bases and add the exponents)
Hi Rock, The rule you have written is incorrect. if you multiply bases, you will get (9x^2)^4. see. 9 is a perfect square (3 * 3= 9). and x^4 is a perfect square ( x^2 * x^2). so 9x^4 can be broken into two parts ---> 3x^2 * 3x^2 Rock, just a suggestion, give time to this concept of indices. Do as many problems as you can. Aditya
|
|
| |
|
|
|
rockrock
|
Post subject: Re: Simplifying exponents Posted: Tue Aug 03, 2010 9:23 am |
|
|
| Course Students |
|
|
Posts: 38
|
|
Ok Let me rephrase this a bit...
The example I used was 2^2 * 2^2 = (2^2)^2 which is 2^4. Here, the same base/add exponent rule applies, correct?
But with 3x^2 * 3x^2 -- the result is 9x^4, not 3x^4...., we dont retain the 3x base as I did in the above example. Is it because there are two integers 3 and x - that we don't retain the base?
Or is the rule that if you have same base/same exponent you are supposed to square them and not apply the add exponent rules.
so
step 1: 3^2 * 3^2 =
step 2: (3^2)^2 =
step 3: 3^4.
You cant skip the 2nd step when another variable (x) is involved.
|
|
| |
|
|
|
adiagr
|
Post subject: Re: Simplifying exponents Posted: Tue Aug 03, 2010 1:38 pm |
|
|
| Students |
|
|
Posts: 89
|
rockrock wrote: Ok Let me rephrase this a bit...
The example I used was 2^2 * 2^2 = (2^2)^2 which is 2^4. Here, the same base/add exponent rule applies, correct?
But with 3x^2 * 3x^2 -- the result is 9x^4, not 3x^4...., we dont retain the 3x base as I did in the above example. Is it because there are two integers 3 and x - that we don't retain the base?
Or is the rule that if you have same base/same exponent you are supposed to square them and not apply the add exponent rules.
so
step 1: 3^2 * 3^2 =
step 2: (3^2)^2 =
step 3: 3^4.
You cant skip the 2nd step when another variable (x) is involved. Ok. it is -->3* (x^2) and not (3x)^2. If it were --> (3x)^2 * (3x)^2 Then same base add exponent rule would have applied and answer would have been: (3x)^4 so returning back to 3* (x^2)we can write [3*(x^2)] * [3*(x^2)] as: (3 * 3) * (x^2 * x^2)Now in (3 * 3), base is 3, exponent is 1 (add exponents, you get 2) => 3^2In (x^2 * x^2)base is x, exponent is 2 (add exponents, you get 4) => x^43^2 = 9 so combined expression becomes 9*x^4. Aditya
|
|
| |
|
|
|
mschwrtz
|
Post subject: Re: Simplifying exponents Posted: Thu Sep 02, 2010 5:10 pm |
|
|
| ManhattanGMAT Staff |
|
|
Posts: 506
|
|
I still get a bit confused because when I use an example of same base same exponent it doesn't add up.
If 9x^4 ---> 3x^2 * 3x^2 (multiply bases and add the exponents)
then
same base,same exponent should also mean:
2^2 * 2^2 = 4^4.... (multiply the bases and add the exponents)
but instead the answer is 4^2 (retain the exponent, multiply the bases).
Where am I going wrong here?
It's not "multiply bases and add the exponents." The bases are x, not 9, 3, etc. Those constants in the original problem are coefficients, not bases. In your new problem, the constants 2 and 4 are bases rather than coefficients.
Here's a very general account:
(ax^b)^c=(a^c)(x^(bc)
That is, you raise the coefficient to the power c, and raise the base to the product of the powers bc.
However, you'd be better of keeping this less general. Why not just expand it into a multiplication question?
By the way, it was very smart of you to consider simple counter-examples here. Properties of exponents can be a lot more accessible/comprehensible with small constants than with large constants or with variables.
|
|
| |
|
|
|
