P and q are positive numbers such that `p^q =q^p and q =9p`. The value of p is

Options

A. `sqrt9`
B. `root6(9)`
C. `root2(9)`
D. `root8(9)`

Answer

To solve the equation \( p^q = q^p \) given that \( q = 9p \), we can follow these steps:

### Step 1: Substitute \( q \) in the equation
We know that \( q = 9p \). We can substitute this into the equation \( p^q = q^p \):
\[ p^{9p} = (9p)^p \]

### Step 2: Expand the right-hand side
Now, we can expand the right-hand side:
\[ (9p)^p = 9^p \cdot p^p \]
So, the equation becomes:
\[ p^{9p} = 9^p \cdot p^p \]

### Step 3: Rearrange the equation
We can rearrange the equation to isolate terms involving \( p \):
\[ p^{9p} = 9^p \cdot p^p \implies \frac{p^{9p}}{p^p} = 9^p \]
This simplifies to:
\[ p^{8p} = 9^p \]

### Step 4: Take the logarithm of both sides
Taking the logarithm of both sides gives us:
\[ \log(p^{8p}) = \log(9^p) \]
Using the properties of logarithms, we can rewrite this as:
\[ 8p \log p = p \log 9 \]

### Step 5: Divide both sides by \( p \) (since \( p > 0 \))
We can divide both sides by \( p \) (since \( p \) is positive):
\[ 8 \log p = \log 9 \]

### Step 6: Solve for \( \log p \)
Now, we can solve for \( \log p \):
\[ \log p = \frac{\log 9}{8} \]

### Step 7: Exponentiate to find \( p \)
Exponentiating both sides, we find:
\[ p = 10^{\frac{\log 9}{8}} = 9^{\frac{1}{8}} = \sqrt[8]{9} \]

### Step 8: Express in terms of roots
We can express \( \sqrt[8]{9} \) as:
\[ p = \sqrt{9^{\frac{1}{4}}} = \sqrt{3} \]

Thus, the value of \( p \) is \( \sqrt{9} \).

### Final Answer:
The value of \( p \) is \( \sqrt{9} \).

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