By what number should `((-3)/(2)) ^(-3)` be divided so that the quotient is `((9)/(4)) ^(-2)` ?

Options

A. `(-3)/2`
B. `3/2`
C. `(-2)/3`
D. `2/3`

Answer

To solve the problem step by step, we need to find the number \( x \) such that:

\[ \frac{(-3/2)^{-3}}{x} = (9/4)^{-2} \]

### Step 1: Rewrite the equation
We start by rewriting the equation to isolate \( x \):

\[ x = \frac{(-3/2)^{-3}}{(9/4)^{-2}} \]

### Step 2: Simplify the negative exponents
Using the property of exponents that states \( a^{-n} = \frac{1}{a^n} \), we can rewrite both sides:

\[ x = \frac{1}{(-3/2)^{3}} \cdot (9/4)^{2} \]

### Step 3: Calculate \( (-3/2)^{3} \)
Now, we calculate \( (-3/2)^{3} \):

\[ (-3/2)^{3} = \frac{(-3)^{3}}{(2)^{3}} = \frac{-27}{8} \]

### Step 4: Calculate \( (9/4)^{2} \)
Next, we calculate \( (9/4)^{2} \):

\[ (9/4)^{2} = \frac{9^{2}}{4^{2}} = \frac{81}{16} \]

### Step 5: Substitute back into the equation
Now we substitute these values back into the equation for \( x \):

\[ x = \frac{1}{\left(\frac{-27}{8}\right)} \cdot \left(\frac{81}{16}\right) \]

### Step 6: Simplify the expression
This simplifies to:

\[ x = \frac{81}{16} \cdot \left(-\frac{8}{27}\right) \]

### Step 7: Multiply the fractions
Now, we multiply the fractions:

\[ x = \frac{81 \cdot (-8)}{16 \cdot 27} \]

### Step 8: Simplify the fraction
Calculating the numerator and the denominator:

\[ x = \frac{-648}{432} \]

### Step 9: Reduce the fraction
Now we simplify \( \frac{-648}{432} \):

\[ x = -\frac{648 \div 216}{432 \div 216} = -\frac{3}{2} \]

### Final Answer
Thus, the number by which \( \left(-\frac{3}{2}\right)^{-3} \) should be divided to obtain \( \left(\frac{9}{4}\right)^{-2} \) is:

\[ \boxed{-\frac{3}{2}} \]