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We are given the following:

\[
3^{9x} = 64 \quad \text{and} \quad 5^{\sqrt{p}} = 64
\]
We are asked to find the value of the expression:

\[
\frac{\left(3^{x-1}\right)^3}{\sqrt[5]{5^p}}
\]

### Step 1: Solving for \(x\)
We know that:
\[
3^{9x} = 64
\]
Taking the logarithm base 3 on both sides:
\[
9x = \log_3(64)
\]
So:
\[
x = \frac{\log_3(64)}{9}
\]

### Step 2: Solving for \(p\)
We are also given:
\[
5^{\sqrt{p}} = 64
\]
Taking the logarithm base 5 on both sides:
\[
\sqrt{p} = \log_5(64)
\]
Squaring both sides gives:
\[
p = \left(\log_5(64)\right)^2
\]

### Step 3: Simplifying the expression
Now, we need to simplify the expression:
\[
\frac{\left(3^{x-1}\right)^3}{\sqrt[5]{5^p}} = \frac{3^{3(x-1)}}{5^{p/5}}
\]

Substitute the values of \(x\) and \(p\) into this expression to compute the final value.

Let me compute that.The simplified form of the given expression is:

\[
\frac{4}{27} \cdot 5^{-\frac{36 \cdot (\log(2))^2}{5 \cdot (\log(5))^2}}
\]

This represents the value of the expression based on the logarithmic calculations of the given powers. The result is expressed in terms of logarithms for precision without a calculator.

Would you like further clarification on any of these steps or a more detailed numerical approximation?

Here are some related questions for further exploration:
1. How does taking logarithms help in solving exponential equations?
2. Can you explain the role of logarithmic identities in simplifying complex expressions?
3. What are the properties of exponents used to combine powers like \( (3^{x-1})^3 \)?
4. How can we solve for unknowns in exponentials using different logarithmic bases?
5. In what situations would numerical approximation be preferred over symbolic solutions?

**Tip:** When solving exponential equations, logarithms are extremely useful for bringing down exponents and simplifying complex terms.

If 3^{9x} = 64 and 5^√p = 64, WITHOUT the use of a calculator, calculate the value of [ (3^{x-1})^3 ] / 5^√p.

Learn how to solve the exponential equations 3^(9x) = 64 and 5^√p = 64 using logarithmic methods, and calculate the value of a complex expression step-by-step without using a calculator.

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