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The prime factorization of \( 27,783 \) is:

\[
27,783 = 3^4 \times 7^3
\]

Thus, comparing it to the equation \( 27,783 = 2^t \times 3^u \times 5^v \times 7^w \):

- \( t = 0 \) (no factor of 2),
- \( u = 4 \) (since \( 3^4 \) appears),
- \( v = 0 \) (no factor of 5),
- \( w = 3 \) (since \( 7^3 \) appears).

So, the values are:
- \( t = 0 \),
- \( u = 4 \),
- \( v = 0 \),
- \( w = 3 \).

Would you like further details or explanations?

Here are 5 questions to further explore this concept:
1. What is prime factorization and why is it useful?
2. How would you factorize a number like 500?
3. Why does the equation not include factors of 2 or 5 in this case?
4. What are the properties of exponents when multiplying prime factorizations?
5. How does knowing prime factors help in solving divisibility problems?

**Tip**: Always check the prime factorization of a number by trying smaller primes (2, 3, 5, etc.) sequentially to avoid errors.

If 27,783 = 2^t × 3^u × 5^v × 7^w, find the values of t, u, v and w.

Learn how to find the prime factorization of 27,783 and solve for the exponents t, u, v, and w in the equation 27,783 = 2^t × 3^u × 5^v × 7^w. This problem covers key concepts in number theory and exponents, suitable for students in Grades 8-10.

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