Step 1: Understanding the Question:
The question asks us to find the value of the expression \(x + y - z\), where \(x\), \(y\), and \(z\) are the exponents of the prime factors 2, 3, and 5, respectively, in the prime factorization of the number 540.
Step 2: Key Formula or Approach:
The fundamental approach is to perform the prime factorization of 540. This means expressing 540 as a product of its prime factors. Once we have this form, we can compare it to the given expression \(2^x \times 3^y \times 5^z\) to determine the values of \(x\), \(y\), and \(z\).
Step 3: Detailed Explanation:
First, we find the prime factorization of 540 by repeated division:
\[
540 = 10 \times 54
\]
Now, we factorize 10 and 54 further:
\[
10 = 2 \times 5
\]
\[
54 = 2 \times 27 = 2 \times 3 \times 3 \times 3 = 2 \times 3^3
\]
Combining these factors, we get the prime factorization of 540:
\[
540 = (2 \times 5) \times (2 \times 3^3)
\]
\[
540 = 2 \times 2 \times 3^3 \times 5 = 2^2 \times 3^3 \times 5^1
\]
Now, we compare this result with the given equation: \(540 = 2^x \times 3^y \times 5^z\).
By comparing the exponents of the corresponding prime bases, we find:
\[
x = 2, \quad y = 3, \quad z = 1
\]
Finally, we calculate the value of the expression \(x + y - z\):
\[
x + y - z = 2 + 3 - 1 = 4
\]
Step 4: Final Answer:
The value of \(x + y - z\) is 4.
\[
\boxed{4}
\]