find the trailing zero in (20^5)*(50^25)*(75^30)

To find the number of trailing zeros in the expression \( (20^5) \times (50^25) \times (75^30) \), we need to consider the number of factors of 5 and 2 in the factorization of this expression, because a trailing zero is produced by a factor of 10, which is \( 2 \times 5 \).

### Step 1: Prime factorization
Let's factorize each term into its prime factors:
- \( 20 = 2^2 \times 5 \)
- \( 50 = 2 \times 5^2 \)
- \( 75 = 3 \times 5^2 \)

Now, raise these factorizations to their respective powers:
- \( 20^5 = (2^2 \times 5)^5 = 2^{10} \times 5^5 \)
- \( 50^{25} = (2 \times 5^2)^{25} = 2^{25} \times 5^{50} \)
- \( 75^{30} = (3 \times 5^2)^{30} = 3^{30} \times 5^{60} \)

### Step 2: Combine the factorizations
Now multiply the factorizations together:
\[
(20^5) \times (50^{25}) \times (75^{30}) = 2^{10} \times 5^5 \times 2^{25} \times 5^{50} \times 3^{30} \times 5^{60}
\]
Combining the powers of the same bases:
\[
= 2^{10 + 25} \times 5^{5 + 50 + 60} \times 3^{30}
\]
\[
= 2^{35} \times 5^{115} \times 3^{30}
\]

### Step 3: Find the number of trailing zeros
The number of trailing zeros is determined by the minimum power of 2 and 5 in the factorization. In this case:
- The power of 2 is \( 35 \).
- The power of 5 is \( 115 \).

The number of trailing zeros is determined by the lesser of these two values.

\[
\text{Number of trailing zeros} = \min(35, 115) = 35
\]

### Final Answer:
The number of trailing zeros in the expression \( (20^5) \times (50^{25}) \times (75^{30}) \) is **35**.

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:
1. How do you find the number of trailing zeros in a factorial like \(100!\)?
2. What is the significance of prime factorization in finding trailing zeros?
3. How can you calculate trailing zeros in large powers like \(1000^{1000}\)?
4. How does the distribution of prime factors affect the trailing zeros?
5. Can you determine the trailing zeros in a product of factorials, e.g., \(10! \times 15!\)?
6. What is the maximum number of trailing zeros a number can have with given factors?
7. How does the presence of factors like 3 affect trailing zeros in products?
8. What are the trailing zeros in combinations like \( \binom{100}{50} \)?

**Tip:** To quickly estimate trailing zeros in large factorials, focus on counting multiples of 5.

Discover how to find the number of trailing zeros in the expression (20^5)*(50^25)*(75^30) using prime factorization and exponentiation. Learn about the importance of factors of 2 and 5 in determining trailing zeros. Suitable for advanced high school students and beyond.

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