https://file.aiquickdraw.com/mathbot/file/1726154571732p8p52h13.jpg

The equation you provided is:

\[
(17)^{2.5} \times (17)^x = 83,521
\]

### Solution:

1. Combine the exponents of 17 using the properties of exponents:

\[
(17)^{2.5} \times (17)^x = 17^{2.5 + x}
\]

This simplifies to:

\[
17^{2.5 + x} = 83,521
\]

2. Now express 83,521 as a power of 17. By calculating, we find:

\[
83,521 = 17^6
\]

3. So, we equate the exponents:

\[
17^{2.5 + x} = 17^6
\]

This gives us the equation:

\[
2.5 + x = 6
\]

4. Solve for \(x\):

\[
x = 6 - 2.5 = 3.5
\]

### Final Answer:
\[
x = 3.5
\]

Let me know if you would like further clarification or more details!

Here are 5 questions related to this topic:

1. What are the properties of exponents used to simplify this equation?
2. How do you express a number as a power of another number?
3. Can you solve a similar equation with a different base?
4. How would the solution change if the base was not a perfect power?
5. How do logarithms relate to solving this type of equation?

**Tip:** When working with exponents, remember that adding and subtracting exponents is straightforward, but multiplying or dividing often requires using logarithms.

Learn how to solve the exponential equation (17)^2.5 x (17)^x = 83,521 by combining exponents and determining the unknown value of x. This problem covers algebraic concepts related to exponents and logarithms, suitable for students in Grades 9-12.

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