By what number should `((2)/(3))^3` be divided so that the quotient is `((4)/(27))^2`?

Options

A. `(27)/(2)`
B. `(9)/(2)`
C. `(2)/(27)`
D. `(2)/(9)`

Answer

To solve the problem step by step, we need to find the number by which \(\left(\frac{2}{3}\right)^3\) should be divided so that the quotient is \(\left(\frac{4}{27}\right)^2\).

### Step 1: Write the equation
We start by setting up the equation based on the problem statement:

\[ \frac{\left(\frac{2}{3}\right)^3}{x} = \left(\frac{4}{27}\right)^2 \]

### Step 2: Calculate \(\left(\frac{2}{3}\right)^3\)
Next, we calculate \(\left(\frac{2}{3}\right)^3\):

\[ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \]

### Step 3: Calculate \(\left(\frac{4}{27}\right)^2\)
Now we calculate \(\left(\frac{4}{27}\right)^2\):

\[ \left(\frac{4}{27}\right)^2 = \frac{4^2}{27^2} = \frac{16}{729} \]

### Step 4: Substitute back into the equation
Now we substitute these values back into our equation:

\[ \frac{\frac{8}{27}}{x} = \frac{16}{729} \]

### Step 5: Cross-multiply to solve for \(x\)
Cross-multiplying gives us:

\[ 8 \cdot 729 = 16 \cdot 27 \cdot x \]

Calculating \(8 \cdot 729\):

\[ 8 \cdot 729 = 5832 \]

Calculating \(16 \cdot 27\):

\[ 16 \cdot 27 = 432 \]

So the equation becomes:

\[ 5832 = 432x \]

### Step 6: Solve for \(x\)
Now, we solve for \(x\):

\[ x = \frac{5832}{432} \]

Calculating \(x\):

\[ x = 13.5 \]

### Step 7: Final answer
Thus, the number by which \(\left(\frac{2}{3}\right)^3\) should be divided is:

\[ x = 13.5 \]