The value of `sqrt(P^3sqrt(P^3 sqrt(P^3)))` is -

Options

A. `P^(21//8)`
B. `P^(27//8)`
C. `P^(1//27)`
D. `P^(8//21)`

Answer

To solve the expression \( \sqrt{P^3 \sqrt{P^3 \sqrt{P^3}}} \), we will simplify it step by step.

### Step 1: Simplify the innermost square root
Start with the innermost expression:
\[ \sqrt{P^3} \]
This can be rewritten using the exponent rule:
\[ \sqrt{P^3} = P^{3/2} \]

### Step 2: Substitute back into the expression
Now substitute this back into the original expression:
\[ \sqrt{P^3 \sqrt{P^3}} = \sqrt{P^3 \cdot P^{3/2}} \]

### Step 3: Combine the exponents
Using the property of exponents (adding exponents when multiplying with the same base):
\[ P^3 \cdot P^{3/2} = P^{3 + 3/2} \]
To add these exponents, convert \(3\) to a fraction:
\[ 3 = \frac{6}{2} \]
So,
\[ 3 + \frac{3}{2} = \frac{6}{2} + \frac{3}{2} = \frac{9}{2} \]
Thus, we have:
\[ \sqrt{P^3 \cdot P^{3/2}} = \sqrt{P^{9/2}} \]

### Step 4: Simplify the square root
Now simplify the square root:
\[ \sqrt{P^{9/2}} = P^{(9/2) \cdot (1/2)} = P^{9/4} \]

### Step 5: Substitute back into the outer expression
Now substitute this back into the outer expression:
\[ \sqrt{P^3 \cdot P^{9/4}} = \sqrt{P^{3 + 9/4}} \]
Convert \(3\) to a fraction:
\[ 3 = \frac{12}{4} \]
So,
\[ 3 + \frac{9}{4} = \frac{12}{4} + \frac{9}{4} = \frac{21}{4} \]
Thus, we have:
\[ \sqrt{P^{3 + 9/4}} = \sqrt{P^{21/4}} \]

### Step 6: Final simplification
Now simplify the square root:
\[ \sqrt{P^{21/4}} = P^{(21/4) \cdot (1/2)} = P^{21/8} \]

### Final Answer
The value of \( \sqrt{P^3 \sqrt{P^3 \sqrt{P^3}}} \) is:
\[ P^{21/8} \]