Find the cube of `7(2)/5`

Options

A. `50653/125`
B. `50652/125`
C. `50659/125`
D. `50658/125`

Answer

To find the cube of \(7 \frac{2}{5}\), we can follow these steps:

### Step 1: Convert the mixed number to an improper fraction.
First, we convert \(7 \frac{2}{5}\) into an improper fraction.

\[ 7 \frac{2}{5} = \frac{(7 \times 5) + 2}{5} = \frac{35 + 2}{5} = \frac{37}{5} \]

### Step 2: Find the cube of the improper fraction.
Now we need to find the cube of \(\frac{37}{5}\).

\[ \left( \frac{37}{5} \right)^3 = \frac{37^3}{5^3} \]

### Step 3: Calculate \(37^3\).
To find \(37^3\), we can first calculate \(37^2\) and then multiply it by \(37\).

\[ 37^2 = 37 \times 37 \]

Using the column method or direct multiplication:

\[ 37 \times 37 = 1369 \]

Now, we multiply \(1369\) by \(37\):

\[ 37^3 = 1369 \times 37 \]

Calculating \(1369 \times 37\):

\[ 1369 \times 37 = 1369 \times (30 + 7) = 1369 \times 30 + 1369 \times 7 \]

Calculating each part:

\[ 1369 \times 30 = 41070 \]
\[ 1369 \times 7 = 9583 \]

Adding these two results together:

\[ 41070 + 9583 = 50653 \]

So, \(37^3 = 50653\).

### Step 4: Calculate \(5^3\).
Now we calculate \(5^3\):

\[ 5^3 = 5 \times 5 \times 5 = 125 \]

### Step 5: Combine the results.
Now we can combine the results from steps 3 and 4:

\[ \left( \frac{37}{5} \right)^3 = \frac{37^3}{5^3} = \frac{50653}{125} \]

### Final Answer:
Thus, the cube of \(7 \frac{2}{5}\) is:

\[ \frac{50653}{125} \]