Question about simple algebraic exponential property

[zell_D]

I haven't been taking math for 3 years so I have a question about the following:

is 14^m6n2 the same as 2^mn x 7^m5n

basically asking this because I am not sure whether or not I can factor the terms out like this

if this information is insufficient I can post the whole problem. Thanks in advance.

 

[songoku]

It's not the same

$$14^{m^6 n^2} = (2 x 7)^{m^6 n^2} = 2^{m^6 n^2} 7^{m^6 n^2}$$

 

[zell_D]

ok so then i do not know how i would arrive at the right answer:

14^m6n2 / 2^mn

i know what the right answer is, and i know how they did it. but i can't seem to grasp the reasoning

PS: the answer is 7^m5n

I was under the impression that you can only subtract the powers through the same base, but apparently i was wrong over here?

or are numbers different than variables?

 

[Kaimyn]

They behave in the same manner, but you know more properties of them. A variable can be any number.

For example, "songoku" manipulated 14 to read (2*7). You probably already know that $$(ab)^{n}=a^{n}b^{n}$$. Thus, we can write:

$$\frac{14^{m^{6}n^{2}}}{2^{mn}}$$ = $$\frac{(2*7)^{m^{6}n^{2}}}{2^{mn}}=\frac{2^{m^{6}n^{2}}*7^{m^{6}n^{2}}}{2^{mn}}}$$.

Is this equal to $$7^{m^{5}n}$$?

 

[zell_D]

i know with the same base i can reduce, but i don't get 7^m5n

 

[songoku]

maybe you can post the whole question?

 

[zell_D]

i did:
its

14m6n2 / 2mn

and the answer being 7m5n

 

[songoku]

If so, the answer is wrong

Just check it : let m = 1 and n = 2

$$\frac{14^{m^6 n^2}}{2^{mn}} = \frac{14^4}{2^2} = 9604$$

$$7^{m^5n} = 7^2 = 49$$

 

[Bohrok]

So it looks like you really have $$\frac{14m^6} {2mn}$$ and the simplified answer is $$\frac{7m^5} {n}$$

This should be a lot easier for you to do than what you were doing.

 

[zell_D]

only thing i can guess is that the book is wrong, the number substitution proves this