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

Options

A. `((-1)/(27))`
B. `((-2)/(27))`
C. `((2)/(27))`
D. `((1)/(27))`

Answer

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

### Step 1: Write the equation
Let \( x \) be the number we need to find. According to the problem, we have:
\[ \frac{(-2/3)^{-3}}{x} = (4/27)^{-2} \]

### Step 2: Rewrite the equation to solve for \( x \)
We can rearrange the equation to isolate \( x \):
\[ x = \frac{(-2/3)^{-3}}{(4/27)^{-2}} \]

### Step 3: Simplify the negative exponents
Recall that \( a^{-m} = \frac{1}{a^m} \). Therefore, we can rewrite the terms with negative exponents:
\[ (-2/3)^{-3} = \frac{1}{(-2/3)^3} \quad \text{and} \quad (4/27)^{-2} = \frac{1}{(4/27)^2} \]

### Step 4: Substitute back into the equation
Substituting these into our expression for \( x \):
\[ x = \frac{\frac{1}{(-2/3)^3}}{\frac{1}{(4/27)^2}} = \frac{(4/27)^2}{(-2/3)^3} \]

### Step 5: Calculate \((4/27)^2\) and \((-2/3)^3\)
Now we compute each part:
\[ (4/27)^2 = \frac{4^2}{27^2} = \frac{16}{729} \]
\[ (-2/3)^3 = \frac{(-2)^3}{3^3} = \frac{-8}{27} \]

### Step 6: Substitute back into the equation
Now substituting these values back into the expression for \( x \):
\[ x = \frac{\frac{16}{729}}{\frac{-8}{27}} = \frac{16}{729} \times \frac{27}{-8} \]

### Step 7: Simplify the fraction
Now we simplify:
\[ x = \frac{16 \times 27}{729 \times -8} \]
Calculating \( 729 = 27^2 \), we can simplify:
\[ x = \frac{16 \times 27}{27^2 \times -8} = \frac{16}{27 \times -8} = \frac{16}{-216} = \frac{-2}{27} \]

### Final Answer
Thus, the number by which \((-2/3)^{-3}\) should be divided is:
\[ \boxed{\frac{-2}{27}} \]