Find the value of `(3^(2))^(5log_(3)x)`:

Options

A. `10^(x)`
B. `x^(4)`
C. 10x
D. `x^(10)`

Answer

To find the value of \( (3^2)^{5 \log_3 x} \), we can follow these steps:

### Step 1: Simplify the expression using the power of a power property
Using the property of exponents that states \( (a^m)^n = a^{m \cdot n} \), we can rewrite the expression:

\[ (3^2)^{5 \log_3 x} = 3^{2 \cdot (5 \log_3 x)} \]

### Step 2: Multiply the exponents
Now, we simplify the exponent:

\[ 2 \cdot (5 \log_3 x) = 10 \log_3 x \]

So, we can rewrite the expression as:

\[ 3^{10 \log_3 x} \]

### Step 3: Apply the property of logarithms
Using the property of logarithms that states \( a^{\log_a b} = b \), we can simplify further:

\[ 3^{10 \log_3 x} = (3^{\log_3 x})^{10} \]

### Step 4: Simplify \( 3^{\log_3 x} \)
From the property mentioned above, we know that \( 3^{\log_3 x} = x \). Therefore, we have:

\[ (3^{\log_3 x})^{10} = x^{10} \]

### Final Result
Thus, the value of \( (3^2)^{5 \log_3 x} \) is:

\[ x^{10} \]