ln the following questions in the place of question mark which sign would be prefered.
[{32(512)^(1/3)}div(16)] ? sqrt256

Options

A. `gt` (greater than)
B. `=`(equal to)
C. `lt` (less than)
D. `ge`(either greater than or equal to)

Answer

To solve the problem step by step, we need to evaluate the expression on the left side and compare it with the square root of 256 on the right side.

### Step 1: Evaluate the left side expression
The left side expression is given as:
\[ \frac{32 \times (512)^{1/3}}{16} \]

### Step 2: Calculate \( (512)^{1/3} \)
First, we need to find the cube root of 512.
\[ 512 = 8^3 \quad \text{(since \(8 \times 8 \times 8 = 512\))} \]
Thus,
\[ (512)^{1/3} = 8 \]

### Step 3: Substitute back into the expression
Now substitute \( (512)^{1/3} \) back into the left side expression:
\[ \frac{32 \times 8}{16} \]

### Step 4: Multiply \(32\) and \(8\)
Now, we calculate:
\[ 32 \times 8 = 256 \]

### Step 5: Divide by \(16\)
Now, we divide \(256\) by \(16\):
\[ \frac{256}{16} = 16 \]

### Step 6: Evaluate the right side expression
Now we evaluate the right side expression, which is:
\[ \sqrt{256} \]
Calculating the square root:
\[ \sqrt{256} = 16 \quad \text{(since \(16 \times 16 = 256\))} \]

### Step 7: Compare both sides
Now we have:
Left side: \(16\)
Right side: \(16\)

### Conclusion
Since both sides are equal, we can conclude:
\[ 16 = 16 \]
Thus, we use the equals sign \(=\) in place of the question mark.

### Final Answer
The preferred sign is \(=\).

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