Prime factorization, Exponents

[turdferguson]

This was taken from a math contest a few months ago.

Homework Statement

x^x*y^y=z^z

find z if:
x=2⁸ * 3⁸
y=2¹² * 3⁶

Homework Equations

Theres undoubtably some trick, but I have yet to find it

The Attempt at a Solution

Dont even think about calculator

I showed my math teacher, and he was able to find that z^z=2^(211*37*11) * 3^(211*37*7)

or 2^49268736 * 3^31352832

How do you get z alone?

 

[Werg22]

[tex]z^{z} = 2^{2^{11}*3^{7}*11} * 3^{2^{11}*3^{7}*7} = ({2^{11}})^{2^{11}*3^{7}}*({3^{7}})^{2^{11}*3^{7}} = ({2^{11}*3^{7}})^{2^{11}*3^{7}} [/tex]

[tex]z = {2^{11}*3^{7}} [/tex]

 

[Architeuthis Dux]

I would approach this problem by first recognizing that prime factorization and exponents are essential mathematical concepts that allow us to break down and simplify complex equations. In this problem, we are given two equations with variables x and y, and we are asked to find the value of z. To do this, we can use the properties of exponents and prime factorization to simplify the equations and solve for z.

First, we can rewrite the equations using exponents to make them easier to work with: xx*yy = zz can be written as (28^2)*(38^2) = z^2 and (212^2)*(36^2) = z^2. This allows us to see that z is the square root of the product of the two equations. Using the properties of exponents, we can also simplify the equations to (2^2*7^2)*(2^3*3^3) = z^2 and (2^2*3^2*53^2)*(2^2*3^3) = z^2.

Next, we can use prime factorization to further simplify the equations. The prime factors of 28 are 2 and 7, and the prime factors of 38 are 2 and 19. Similarly, the prime factors of 212 are 2, 3, and 53, and the prime factors of 36 are 2, 3, and 3. This allows us to rewrite the equations as (2^2*7^2)*(2^3*3^3) = z^2 and (2^2*3^2*53^2)*(2^2*3^3) = z^2.

Now, we can combine like terms and use the properties of exponents to simplify the equations even further. This results in (2^5*7^2*3^3) = z^2 and (2^4*3^5*53^2) = z^2. Taking the square root of both sides, we get z = 2^2*3^2*7*53 = 249268736.

Therefore, the value of z is 249268736. By using the concepts of prime factorization and exponents, we were able to simplify the equations and find the value of z without using a calculator. This is a great example of how these mathematical concepts can be