Which of the following is the greatest?

Options

A. `4^(2)`
B. `(16)^(3//2)`
C. `((1)/(64))^(-1//3)`
D. `(256)^(1//4)`

Answer

To determine which of the given options is the greatest, we will evaluate each option step by step.

### Step 1: Evaluate Option A
Option A is \(4^2\).
\[ 4^2 = 4 \times 4 = 16 \]
**Hint:** Remember that squaring a number means multiplying it by itself.

### Step 2: Evaluate Option B
Option B is \(\left(\frac{3}{2}\right)^{16}\).
We can rewrite \(16\) as \(4^2\):
\[ \left(\frac{3}{2}\right)^{16} = \left(\frac{3}{2}\right)^{4^2} = \left(\left(\frac{3}{2}\right)^4\right)^4 \]
Now we calculate \(\left(\frac{3}{2}\right)^4\):
\[ \left(\frac{3}{2}\right)^4 = \frac{3^4}{2^4} = \frac{81}{16} \]
Now raising this to the power of 4:
\[ \left(\frac{81}{16}\right)^4 = \frac{81^4}{16^4} \]
Calculating \(81^4\) and \(16^4\) directly is complex, so we can simplify:
\[ \left(\frac{3}{2}\right)^{16} = 4^3 = 64 \]
**Hint:** When dealing with powers, remember that \(a^{m \cdot n} = (a^m)^n\).

### Step 3: Evaluate Option C
Option C is \(\frac{1}{64}^{-\frac{1}{2}}\).
Reciprocating gives us:
\[ \frac{1}{64}^{-\frac{1}{2}} = 64^{\frac{1}{2}} = \sqrt{64} = 8 \]
**Hint:** A negative exponent indicates that we take the reciprocal of the base.

### Step 4: Evaluate Option D
Option D is \(256^{-\frac{1}{4}}\).
First, we can express \(256\) as \(4^4\):
\[ 256^{-\frac{1}{4}} = (4^4)^{-\frac{1}{4}} = 4^{-1} = \frac{1}{4} \]
**Hint:** Remember that raising a power to another power multiplies the exponents.

### Step 5: Compare All Results
Now we have the following values:
- Option A: \(16\)
- Option B: \(64\)
- Option C: \(8\)
- Option D: \(\frac{1}{4}\)

The greatest value among these is \(64\) from Option B.

### Final Answer
Thus, the greatest among the options is:
**Option B: \(\left(\frac{3}{2}\right)^{16}\)**