If a and b are whole number such that `a^(b)=512` where a `gt` b and 1 `lt` b `lt` 4 then `bsqrt(a)`=

Options

A. 2
B. 3
C. 4
D. cannot be determined

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) such that \( a^b = 512 \), with the conditions that \( a > b \) and \( 1 < b < 4 \).

### Step 1: Identify possible values for \( b \)

Since \( b \) must be a whole number and \( 1 < b < 4 \), the possible values for \( b \) are 2 and 3.

### Step 2: Calculate \( a \) for each possible value of \( b \)

1. **If \( b = 2 \)**:
\[ a^2 = 512 \implies a = \sqrt{512} = \sqrt{256 \times 2} = 16\sqrt{2} \]
Since \( a \) must be a whole number, this value of \( b \) is not valid.

2. **If \( b = 3 \)**:
\[ a^3 = 512 \implies a = \sqrt[3]{512} = 8 \]
Here, \( a = 8 \) is a whole number.

### Step 3: Verify the conditions

- Check if \( a > b \):
\[ 8 > 3 \quad \text{(True)} \]
- Check if \( 1 < b < 4 \):
\[ 1 < 3 < 4 \quad \text{(True)} \]

Both conditions are satisfied with \( a = 8 \) and \( b = 3 \).

### Step 4: Calculate \( b \sqrt{a} \)

Now we need to find \( b \sqrt{a} \):
\[ b \sqrt{a} = 3 \sqrt{8} \]
We can simplify \( \sqrt{8} \):
\[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
Thus,
\[ b \sqrt{a} = 3 \times 2\sqrt{2} = 6\sqrt{2} \]

### Final Answer:
\[ b \sqrt{a} = 6\sqrt{2} \]

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