The value of `(243)^(0.16)xx(243)^(0.04)` is equal to :

Options

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

Answer

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

### Step 1: Identify the Base and Add the Exponents
We notice that both terms have the same base, which is 243. According to the properties of exponents, when multiplying two powers with the same base, we can add the exponents:

\[ (243)^{0.16} \times (243)^{0.04} = (243)^{0.16 + 0.04} \]

### Step 2: Calculate the Sum of the Exponents
Now, we add the exponents:

\[ 0.16 + 0.04 = 0.20 \]

So we can rewrite the expression as:

\[ (243)^{0.20} \]

### Step 3: Rewrite 243 in Terms of a Base
Next, we can express 243 as a power of 3. We know that:

\[ 243 = 3^5 \]

### Step 4: Substitute and Simplify
Now we can substitute \(243\) with \(3^5\) in our expression:

\[ (3^5)^{0.20} \]

### Step 5: Apply the Power of a Power Rule
Using the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\), we can simplify further:

\[ (3^5)^{0.20} = 3^{5 \times 0.20} = 3^{1} \]

### Step 6: Calculate the Final Value
Finally, we calculate \(3^{1}\):

\[ 3^{1} = 3 \]

Thus, the value of \((243)^{0.16} \times (243)^{0.04}\) is equal to **3**.

### Final Answer
The answer is **3**.

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