# Prime factorization, Exponents

• turdferguson
In summary, prime factorization is the process of breaking down a composite number into its prime factors, which are numbers that can only be divided by 1 and themselves. The prime factors of a number can be found using a factor tree or division method. An exponent is a mathematical notation that indicates how many times a number is multiplied by itself, and expressions with exponents can be simplified using the properties of exponents. A prime factor is a prime number that is a factor of a given number, while a prime exponent is the number of times a prime number is used as a factor in an exponent expression.
turdferguson
This was taken from a math contest a few months ago.

xx*yy=zz

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 zz=2(211*37*11) * 3(211*37*7)

or 249268736 * 331352832

How do you get z alone?

$$z^{z} = 2^{2^{11}*3^{7}*11} * 3^{2^{11}*3^{7}*7} = ({2^{11}})^{2^{11}*3^{7}}*({3^{7}})^{2^{11}*3^{7}} = ({2^{11}*3^{7}})^{2^{11}*3^{7}}$$

$$z = {2^{11}*3^{7}}$$

Last edited:

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

## 1. What is prime factorization?

Prime factorization is the process of breaking down a composite number into its prime factors, which are the numbers that can only be divided by 1 and themselves. This process is important in mathematics and is used in various applications, such as simplifying fractions and finding the greatest common factor of two numbers.

## 2. How do you find the prime factors of a number?

To find the prime factors of a number, you can use a factor tree or division method. The factor tree involves continuously dividing the number by its prime factors until all the branches contain only prime numbers. The division method involves dividing the number by the smallest prime number that can evenly divide into it, and then continuing to divide the resulting quotient until it is a prime number.

## 3. What is an exponent?

An exponent is a mathematical notation that indicates the number of times a number (base) is multiplied by itself. It is written as a small number above and to the right of the base number. For example, in 23, 2 is the base and 3 is the exponent. This can also be read as 2 to the power of 3.

## 4. How do you simplify expressions with exponents?

To simplify expressions with exponents, you can use the properties of exponents. These include the product rule (am * an = am+n), the quotient rule (am / an = am-n), and the power rule ((am)n = am*n). These rules allow you to combine and manipulate terms with exponents to simplify the expression.

## 5. What is the difference between a prime factor and a prime exponent?

A prime factor is a prime number that is a factor of a given number. For example, the prime factors of 24 are 2, 2, 2, and 3. A prime exponent, on the other hand, is the number of times a prime number is used as a factor in an exponent expression. For example, in 23, the prime exponent is 3 because 2 is used as a factor 3 times.

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