Find the value of p in the following:
`((4)/(3))^(3) div ((-4)/(3))^(3)=(-1)^(3p)`

Answer

To find the value of \( p \) in the equation

\[ \frac{(4/3)^3}{(-4/3)^3} = (-1)^{3p}, \]

we will follow these steps:

### Step 1: Rewrite the left side using properties of exponents
We can rewrite the left side of the equation using the property of exponents that states \(\frac{a^n}{b^n} = \frac{a}{b}^n\). Therefore, we have:

\[ \frac{(4/3)^3}{(-4/3)^3} = \left(\frac{4}{-4}\right)^3 = \left(-1\right)^3. \]

### Step 2: Simplify the left side
Now, simplifying the left side gives us:

\[ (-1)^3 = -1. \]

### Step 3: Set the left side equal to the right side
Now we have:

\[ -1 = (-1)^{3p}. \]

### Step 4: Analyze the right side
The expression \((-1)^{3p}\) can be simplified based on the value of \(3p\):
- If \(3p\) is odd, \((-1)^{3p} = -1\).
- If \(3p\) is even, \((-1)^{3p} = 1\).

Since the left side is \(-1\), we conclude that \(3p\) must be odd.

### Step 5: Solve for \(p\)
To find \(p\), we can set \(3p = 1\) (the smallest odd integer). Thus, we have:

\[ p = \frac{1}{3}. \]

### Final Answer
Therefore, the value of \( p \) is

\[ \boxed{\frac{1}{3}}. \]

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