If `p^r . P^(-1) . P^s = (sqrt(p^3))^(2) and p^(3//2).p^r = p^s.p^(-1//2)` then the value of `(r + s)^((r + s))` is :

Options

A. a. `27`
B. b. `64`
C. c.`4`
D. d. None of these

Answer

To solve the given problem, we need to work through the equations step by step.

**Step 1: Simplify the first equation.**
We start with the equation:
\[ p^r \cdot p^{-1} \cdot p^s = \left(\sqrt{p^3}\right)^2 \]

The left-hand side can be simplified using the property of exponents:
\[ p^r \cdot p^{-1} \cdot p^s = p^{r + s - 1} \]

The right-hand side simplifies as follows:
\[ \left(\sqrt{p^3}\right)^2 = (p^{3/2})^2 = p^{3} \]

So we have:
\[ p^{r + s - 1} = p^3 \]

**Step 2: Set the exponents equal to each other.**
Since the bases are the same, we can set the exponents equal:
\[ r + s - 1 = 3 \]

**Step 3: Solve for \( r + s \).**
Adding 1 to both sides gives:
\[ r + s = 4 \]

**Step 4: Simplify the second equation.**
Now we look at the second equation:
\[ p^{3/2} \cdot p^r = p^s \cdot p^{-1/2} \]

Again, we can simplify both sides:
Left-hand side:
\[ p^{3/2 + r} \]

Right-hand side:
\[ p^{s - 1/2} \]

So we have:
\[ p^{3/2 + r} = p^{s - 1/2} \]

**Step 5: Set the exponents equal to each other again.**
Setting the exponents equal gives us:
\[ 3/2 + r = s - 1/2 \]

**Step 6: Rearrange the equation.**
Rearranging this gives:
\[ r - s = -1/2 - 3/2 \]
\[ r - s = -2 \]

**Step 7: Solve the system of equations.**
Now we have a system of two equations:
1. \( r + s = 4 \)
2. \( r - s = -2 \)

We can solve for \( r \) and \( s \) by adding these two equations:
\[ (r + s) + (r - s) = 4 - 2 \]
\[ 2r = 2 \]
\[ r = 1 \]

Now substituting \( r \) back into the first equation:
\[ 1 + s = 4 \]
\[ s = 3 \]

**Step 8: Calculate \( (r + s)^{(r + s)} \).**
Now we can find \( (r + s)^{(r + s)} \):
\[ (r + s) = 4 \]
Thus:
\[ (r + s)^{(r + s)} = 4^4 = 256 \]

So the final answer is:
\[ \boxed{256} \]

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