Multiplying Powers

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Probability

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I need to take this a little further with some feed back please to ensure I understand why some powers in index form cannot be written more concisely.

OK starting with a simple example;

3^4 x 3^5 = 3^9

Now...

2^4 x 3^7 = Cannot be written more concisely.

Now I have;

2^3 x 7 x 2^2 x 7^2 = 2^3+2 x 7 x 7^2 = 2^3+2 x 7^1+2 = 2^5 x 7^3

Now I have again different base units...

9 x 3^5 = 3^3 x 3^5 = 3^3+5 = 3^7

Now I have another with different base units, which I'm told cannot be written more concisely.

3^4 x 5^12 = cannot be written more concisely.

Can I not do this...

3 x 3 x 3 x 3 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 =

3 x 5^4 x 3 x 5^4 3 x 5^4 x 3 x 5^4 =

(3 x 5)^4 = (15)^4

Is that not allowed and if not why?
 
3^4 = 81 and 5^12 = 244140625. Multiplying those together gives you 19775390625.
15^4 = 50625.
They are not even close to being the same.
You are mixing up some concepts with factoring and operation rules here.

It looks like you are attempting this: aX + aY + aZ = a(X + Y + Z). But there is no addition in your problem; it is just straight multiplication, which can be rearranged, but aX times aY times aZ does NOT equal a(XYZ); it equals a^3(XYZ).
 
Sorry for any confusion that I have caused here. Its a learning curve and what I'm trying to establish here is a rule for checking when the base units can be changed from one to another, i.e. say 3 to 5 when presented with powers, so 8 x 5^5 I'd be looking to see if the base unit can be changed.
 
OMG!!

If you had written 2^3 x 7 x 2^2 x 7^2 = 2^3+2 x 7 x 7^2 = 2^3+2 x 7^1+2 = 2^5 x 7^3 in my College Algebra class I would have failed you immediately.

You need parenthesis here! Desperately!

2^3 x 7 x 2^2 x 7^2 = 2^(3 + 2) x 7^(1 + 2)

This is not a (relatively) small matter of something like x / x + 2 where you mean x/(x + 2)... you could probably guess this one easily. This is a gross problem with order of operations. Please take this comment seriously. Parenthesis are your friends!

-Dan
 
I agree and I'd thought about it typing it all out but as you know using a keyboard typing that lot out is time consuming to start with.
 
Probability said:
I agree and I'd thought about it typing it all out but as you know using a keyboard typing that lot out is time consuming to start with.
Click to expand...
I'm assuming, when you post something here, you want others to read it and understand what you've written. Typing fewer characters to save time will guarantee, that you will not achieve your goal.
 
Probability said:
I need to take this a little further with some feed back please to ensure I understand why some powers in index form cannot be written more concisely.

OK starting with a simple example;

3^4 x 3^5 = 3^9

Now...

2^4 x 3^7 = Cannot be written more concisely.

Now I have;

2^3 x 7 x 2^2 x 7^2 = 2^3+2 x 7 x 7^2 = 2^3+2 x 7^1+2 = 2^5 x 7^3

Now I have again different base units...

9 x 3^5 = 3^3 x 3^5 = 3^3+5 = 3^7

Now I have another with different base units, which I'm told cannot be written more concisely.

3^4 x 5^12 = cannot be written more concisely.

Can I not do this...

3 x 3 x 3 x 3 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 =

3 x 5^4 x 3 x 5^4 3 x 5^4 x 3 x 5^4 =

(3 x 5)^4 = (15)^4

Is that not allowed and if not why?
Click to expand...
3^4 x 5^12

=3 x 3 x 3 x 3 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5

=(3x5x5x5)*(3x5x5x5)*(3x5x5x5)

=(3 x 5^3) x (3 x 5^3) x (3 x 5^3) x (3 x 5^3)

(3 x 5^3)^4 = (375)^4

If you are repeatedly multiplying (3x5^3) for 4 times, then you get what you were repeatedly multiplying raised to the 4th power. You are NOT repeatedly multiplying 3x5! You are repeatedly multiplying 3x5^3. So in the end you get (3x5^3)^4 = (3x125)^4 = 375^4
 
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