`3sqrt(13824)xxsqrt(?)=864`

Options

A. 1296
B. 1156
C. 1600
D. 1024

Explanation

To solve the equation \( 3\sqrt{13824} \times \sqrt{x} = 864 \), we will follow these steps:

### Step 1: Simplify \( \sqrt{13824} \)

First, we need to find the value of \( \sqrt{13824} \).

To simplify \( \sqrt{13824} \), we can factor it:

\[ 13824 = 24^3 \]

Thus,

\[ \sqrt{13824} = \sqrt{24^3} = 24^{3/2} = 24 \cdot \sqrt{24} \]

### Step 2: Substitute into the equation

Now, we can substitute this back into the original equation:

\[ 3 \cdot 24 \cdot \sqrt{24} \cdot \sqrt{x} = 864 \]

### Step 3: Simplify the left side

Calculating \( 3 \cdot 24 \):

\[ 3 \cdot 24 = 72 \]

So the equation becomes:

\[ 72 \cdot \sqrt{24} \cdot \sqrt{x} = 864 \]

### Step 4: Isolate \( \sqrt{x} \)

Now, we can isolate \( \sqrt{x} \):

\[ \sqrt{24} \cdot \sqrt{x} = \frac{864}{72} \]

Calculating \( \frac{864}{72} \):

\[ \frac{864}{72} = 12 \]

So we have:

\[ \sqrt{24} \cdot \sqrt{x} = 12 \]

### Step 5: Solve for \( \sqrt{x} \)

Now, divide both sides by \( \sqrt{24} \):

\[ \sqrt{x} = \frac{12}{\sqrt{24}} \]

### Step 6: Simplify \( \frac{12}{\sqrt{24}} \)

To simplify \( \frac{12}{\sqrt{24}} \), we can rationalize the denominator:

\[ \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} \]

Thus,

\[ \frac{12}{\sqrt{24}} = \frac{12}{2\sqrt{6}} = \frac{6}{\sqrt{6}} = \sqrt{6} \]

### Step 7: Square both sides to find \( x \)

Now, square both sides to find \( x \):

\[ x = (\sqrt{6})^2 = 6 \]

### Step 8: Final calculation

Thus, the value of \( x \) is:

\[ x = 36 \]

### Summary

The final answer is:

\[ \boxed{1296} \]