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The problem asks to find the final exponent of \(x\) in the given expression:

\[
M = \frac{\left( \sqrt[3]{x} \cdot \sqrt[3]{x} \cdot \sqrt[3]{x} \cdot \sqrt[3]{x} \cdot \sqrt[3]{x} \right)^{243}}{\left( \sqrt[4]{x} \cdot \sqrt[4]{x} \cdot \sqrt[4]{x} \cdot \sqrt[4]{x} \right)^{256}}
\]

We need to simplify the expression for \(M\) by determining the combined exponent of \(x\) in both the numerator and denominator.

### Numerator:
The numerator consists of the product of five cube roots of \(x\), raised to the power of 243. The cube root of \(x\) is \(x^{1/3}\), so:

\[
\sqrt[3]{x} \cdot \sqrt[3]{x} \cdot \sqrt[3]{x} \cdot \sqrt[3]{x} \cdot \sqrt[3]{x} = \left(x^{1/3}\right)^5 = x^{5/3}
\]

Raising this to the power of 243 gives:

\[
\left(x^{5/3}\right)^{243} = x^{(5/3) \cdot 243} = x^{405}
\]

### Denominator:
The denominator consists of the product of four fourth roots of \(x\), raised to the power of 256. The fourth root of \(x\) is \(x^{1/4}\), so:

\[
\sqrt[4]{x} \cdot \sqrt[4]{x} \cdot \sqrt[4]{x} \cdot \sqrt[4]{x} = \left(x^{1/4}\right)^4 = x^{4/4} = x^1
\]

Raising this to the power of 256 gives:

\[
\left(x^{1}\right)^{256} = x^{256}
\]

### Simplifying \(M\):
Now, we divide the two results:

\[
M = \frac{x^{405}}{x^{256}} = x^{405 - 256} = x^{149}
\]

Thus, the final exponent of \(x\) is **149**, which is not in the provided options. I will review my calculations to ensure no simplification errors.Upon reviewing the problem, I see the calculation was correct. The exponent of \(x\) in the final result is indeed \(149\). However, this result does not match any of the provided answer choices (36, 63, 39, 93, 17). 

This may indicate that either there's a typo in the problem's answer key, or the question intends for the exponent to be simplified further, but based on the standard rules of exponents, \(149\) is the accurate result given the expression. 

If you need any further clarification or steps, feel free to ask!

Here are some related questions for further practice:

1. What is the exponent rule for dividing powers with the same base?
2. How do you simplify a product of roots of the same base?
3. What happens when you raise a power to another power?
4. How do you handle exponents when simplifying expressions with both roots and powers?
5. What are common mistakes to avoid when simplifying roots and powers in complex fractions?

**Tip:** Always double-check the answer choices against your final result to catch possible misprints or issues with the options provided.

Find the final exponent of 'x' in the given expression:
M = ((√³x * √³x * √³x * √³x * √³x)⁴²³) / ((√⁴x * √⁴x * √⁴x * √⁴x)²⁵⁶) ; x ≠ 0

Learn how to simplify and find the final exponent of 'x' in a complex expression involving cube roots, fourth roots, and exponentiation. This solution walks through each step, including root and exponent rules, making it ideal for students in grades 10-12.

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