Question

If x=(2^12)(3^6) and y=(2^8)(3^8), what integer z satisfies (x^x)(y^y)=z^z? Write the answer as a product of powers of primes.

Answer 1

z has to satisfy
\[ \large x^xy^y=z^z \]
?

Answer 2

yes

Answer 3

that is
\[ \Large (2^{12}3^6)^{2^{12}3^6}(2^83^8)^{2^83^8}=z^z \]
?

Answer 4

yes

Answer 5

I wonder if logarithms would help here...

Answer 6

I am not sure, I haven't learn log yet

Answer 7

But you're comfortable with basic exponent properties, yes?

Answer 8

yes

Answer 9

Ok, start with what helder_edwin posted and simplify that using rules for exponents and try to get all your bases as prime numbers.

Answer 10

Then you can group the bases together.

Answer 11

can you show me an example?

Answer 12

Alright..

\[\large (2^{12}3^6)^{2^{12}3^6} (2^83^8)^{2^83^8}\]
\[\large \rightarrow 2^{12\cdot2^{12}\cdot3^6}\cdot3^{6\cdot2^{12}\cdot3^6}\cdot 2^{8\cdot2^8\cdot3^8}\cdot3^{8\cdot2^{8}\cdot3^8}\]

Answer 13

With a power-to-a-power, you multiply exponents.

Answer 14

?