Evaluate the following:
`(6 1/2)^3`

Answer

To evaluate \((6 \frac{1}{2})^3\), we can follow these steps:

### Step 1: Convert the mixed number to an improper fraction
First, we need to convert \(6 \frac{1}{2}\) into an improper fraction.

\[ 6 \frac{1}{2} = 6 + \frac{1}{2} = \frac{6 \times 2 + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2} \]

### Step 2: Cube the improper fraction
Next, we will cube the improper fraction \(\frac{13}{2}\).

\[ (6 \frac{1}{2})^3 = \left(\frac{13}{2}\right)^3 = \frac{13^3}{2^3} \]

### Step 3: Calculate \(13^3\) and \(2^3\)
Now, we need to calculate \(13^3\) and \(2^3\).

\[ 2^3 = 8 \]

To calculate \(13^3\), we can use the formula for the cube of a number:

\[ 13^3 = 13 \times 13 \times 13 \]

Calculating \(13 \times 13 = 169\), then:

\[ 169 \times 13 = 1690 + 507 = 2197 \]

So, \(13^3 = 2197\).

### Step 4: Write the result as a fraction
Now we can write the result as:

\[ (6 \frac{1}{2})^3 = \frac{2197}{8} \]

### Final Answer
Thus, the final answer is:

\[ \frac{2197}{8} \]

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