To solve the equation \(\frac{(x-5)^{3/4}}{4} = 27\), follow these steps:

1. **Isolate the expression** involving \(x\):
   \[
   (x-5)^{3/4} = 27 \times 4 = 108
   \]

2. **Raise both sides to the power of \(\frac{4}{3}\)** to eliminate the fractional exponent:
   \[
   x-5 = 108^{\frac{4}{3}}
   \]

3. **Calculate \(108^{\frac{4}{3}}\)**:
   \[
   108^{\frac{4}{3}} = (108^{\frac{1}{3}})^4 = 6^4 = 1296
   \]
   So,
   \[
   x - 5 = 1296
   \]

4. **Solve for \(x\)** by adding 5 to both sides:
   \[
   x = 1296 + 5 = 1301
   \]

Therefore, the solution is \( \boxed{1301} \).

Would you like more details on any step, or do you have any questions?

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Here are five related questions:

1. How would the solution change if the equation was \((x-5)^{3/2} = 27\)?
2. What if the equation was \(\frac{(x+2)^{3/4}}{4} = 16\)?
3. How can you solve \( (2x-3)^{5/2} = 32 \)?
4. What happens if you apply logarithms to both sides of an exponential equation?
5. How do you solve equations involving radicals like \(\sqrt{x-3} = 5\)?

**Tip:** Always check your solution by substituting it back into the original equation.

Learn how to solve the equation (x-5)^(3/4) = 27 with detailed step-by-step instructions. This problem involves exponential equations and fractional exponents, suitable for high school students.

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