Find the value of `(243)^(".l6")xx(243)^(.04)`

Options

A. 0.16
B. `1/3`
C. 3
D. 0.04

Answer

To solve the expression \( (243)^{0.16} \times (243)^{0.04} \), we can follow these steps:

### Step 1: Rewrite the base
First, we recognize that \( 243 \) can be expressed as a power of \( 3 \):
\[ 243 = 3^5 \]

### Step 2: Substitute the base into the expression
Now we can rewrite the original expression using the base \( 3^5 \):
\[ (243)^{0.16} \times (243)^{0.04} = (3^5)^{0.16} \times (3^5)^{0.04} \]

### Step 3: Apply the power of a power rule
Using the power of a power property \( (a^m)^n = a^{m \cdot n} \), we can simplify each term:
\[ (3^5)^{0.16} = 3^{5 \times 0.16} = 3^{0.8} \]
\[ (3^5)^{0.04} = 3^{5 \times 0.04} = 3^{0.2} \]

### Step 4: Combine the exponents
Now we can combine the two terms using the property \( a^m \times a^n = a^{m+n} \):
\[ 3^{0.8} \times 3^{0.2} = 3^{0.8 + 0.2} = 3^{1.0} \]

### Step 5: Simplify the expression
Since \( 3^{1.0} = 3 \), we find that:
\[ (243)^{0.16} \times (243)^{0.04} = 3 \]

### Final Answer
Thus, the value of \( (243)^{0.16} \times (243)^{0.04} \) is:
\[ \boxed{3} \]