Simplifying exponents

[rockrock]

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]

Hello, nice to meet you.

also there is : (x^2)^2=x^(2*2)=x^4

c ya.

[adiagr]

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]

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]

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]

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..

[rockrock]

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]

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]

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]

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)
=> 3^2


In (x^2 * x^2)

base is x, exponent is 2 (add exponents, you get 4)
=> x^4

3^2 = 9

so combined expression becomes 9*x^4.

Aditya

[mschwrtz]

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.