Question

If $540 = 2^x \times 3^y \times 5^z$, find the value of $x + y - z$.

Hint:

To compare exponents:
Always do prime factorization first.
Match bases and compare powers.
This method is common in factorization problems.

Answer 1

Concept: To compare exponents, express the number as a product of its prime factors. Step 1: Prime factorization of 540 \[ 540 = 54 \times 10 \] \[ 54 = 2 \times 27 = 2 \times 3^3 \] \[ 10 = 2 \times 5 \] So, \[ 540 = (2 \times 3^3)(2 \times 5) = 2^2 \times 3^3 \times 5^1 \] Step 2: Compare with given form \[ 540 = 2^x \times 3^y \times 5^z \] Thus, \[ x = 2, \quad y = 3, \quad z = 1 \] Step 3: Find required value \[ x + y - z = 2 + 3 - 1 = 4 \] Final Answer: \[ \boxed{4} \]