What is the real value of `(256)^(0.16)xx(16)^(0.18)` ?

Options

A. 2
B. 4
C. `1/4`
D. `1/2`

Answer

To solve the expression \((256)^{0.16} \times (16)^{0.18}\), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers of 16
We know that:
- \(256 = 16^2\)
- \(16 = 16^1\)

So we can rewrite \(256\) as:
\[ (256)^{0.16} = (16^2)^{0.16} \]

### Step 2: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (16^2)^{0.16} = 16^{2 \cdot 0.16} = 16^{0.32} \]

### Step 3: Rewrite the entire expression
Now, we can rewrite the original expression:
\[ (256)^{0.16} \times (16)^{0.18} = 16^{0.32} \times 16^{0.18} \]

### Step 4: Combine the exponents
Since the bases are the same, we can add the exponents:
\[ 16^{0.32 + 0.18} = 16^{0.5} \]

### Step 5: Simplify the exponent
Now, we can simplify \(16^{0.5}\):
\[ 16^{0.5} = \sqrt{16} \]

### Step 6: Calculate the square root
The square root of \(16\) is:
\[ \sqrt{16} = 4 \]

### Final Answer
Thus, the real value of \((256)^{0.16} \times (16)^{0.18}\) is:
\[ \boxed{4} \]

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