https://file.aiquickdraw.com/mathbot/file/1726039443853b9o7tf0h.jpg

The equation presented in the image is:

\[ 2^x = 5^y = (10)^{-z} \]
and the task is to find:

\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = ?
\]

### Step-by-step Solution:

1. Start by expressing \( 10^{-z} \) in terms of its prime factorization:
   \[
   10 = 2 \times 5 \quad \text{so} \quad 10^{-z} = 2^{-z} \times 5^{-z}.
   \]
   This gives us:
   \[
   2^x = 5^y = 2^{-z} \times 5^{-z}.
   \]

2. Set the first equation \( 2^x = 2^{-z} \), which implies:
   \[
   x = -z.
   \]

3. Similarly, from \( 5^y = 5^{-z} \), we get:
   \[
   y = -z.
   \]

4. Now substitute these results into the expression we need to evaluate:
   \[
   \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
   \]
   Since \( x = -z \) and \( y = -z \), we have:
   \[
   \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{-z} + \frac{1}{-z} + \frac{1}{z}.
   \]

5. Simplifying:
   \[
   \frac{1}{-z} + \frac{1}{-z} + \frac{1}{z} = -\frac{1}{z} - \frac{1}{z} + \frac{1}{z} = -\frac{1}{z}.
   \]

So, the final answer is:

\[
\boxed{0}.
\]

Would you like further clarification on any step or more details? Here are some related questions:

1. How do you derive \( 10^{-z} \) as \( 2^{-z} \times 5^{-z} \)?
2. What is the significance of expressing numbers as powers in such problems?
3. Can you explain why \( x = -z \) and \( y = -z \) hold true for this problem?
4. How would the answer change if the bases of the exponentials were different?
5. Can this problem be solved using logarithms?

**Tip:** Understanding how to manipulate exponents is critical in solving equations involving multiple bases like in this question.

यदि 2^x = 5^y = (10)^{-z}, तो 1/x + 1/y + 1/z = ?

Learn how to solve the exponential equation 2^x = 5^y = (10)^-z and find the value of 1/x + 1/y + 1/z. The problem uses prime factorization and laws of exponents. Ideal for students in Grades 10-12.

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