If `9^(x)=root11(243)`, then what is the value of x ?

Options

A. `(5)/(11)`
B. `(5)/(22)`
C. `(5)/(7)`
D. `(5)/(33)`

Answer

To solve the equation \( 9^{x} = \sqrt[11]{243} \), we can follow these steps:

### Step 1: Rewrite the bases
We know that \( 9 \) can be expressed as \( 3^2 \) and \( 243 \) can be expressed as \( 3^5 \). Thus, we rewrite the equation:

\[ 9^{x} = (3^2)^{x} = 3^{2x} \]

### Step 2: Rewrite the right-hand side
The term \( \sqrt[11]{243} \) can be rewritten using exponents. Since \( 243 = 3^5 \), we have:

\[ \sqrt[11]{243} = (3^5)^{\frac{1}{11}} = 3^{\frac{5}{11}} \]

### Step 3: Set the exponents equal
Now we have:

\[ 3^{2x} = 3^{\frac{5}{11}} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ 2x = \frac{5}{11} \]

### Step 4: Solve for \( x \)
To find \( x \), we divide both sides by 2:

\[ x = \frac{5}{11} \cdot \frac{1}{2} = \frac{5}{22} \]

### Conclusion
Thus, the value of \( x \) is:

\[ \boxed{\frac{5}{22}} \]

---