What is the value of `[(a^(-2)b^(3)) div (a^(1)b^(-1))] xx [(a^(2)b^(-4)) div (a^(-1)b^(2))]`?

Options

A. `b^(2)`
B. `1//b^(2)`
C. `a^(2)`
D. `a^(2)b^(2)`

Answer

To solve the expression \(\left[\frac{a^{-2}b^{3}}{a^{1}b^{-1}}\right] \times \left[\frac{a^{2}b^{-4}}{a^{-1}b^{2}}\right]\), we will simplify each part step by step.

### Step 1: Simplify the first fraction
The first fraction is:
\[ \frac{a^{-2}b^{3}}{a^{1}b^{-1}} \]
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify:
\[ = \frac{a^{-2}}{a^{1}} \cdot \frac{b^{3}}{b^{-1}} = a^{-2-1} \cdot b^{3-(-1)} = a^{-3} \cdot b^{3+1} = a^{-3} \cdot b^{4} \]

### Step 2: Simplify the second fraction
The second fraction is:
\[ \frac{a^{2}b^{-4}}{a^{-1}b^{2}} \]
Again, using the property of exponents:
\[ = \frac{a^{2}}{a^{-1}} \cdot \frac{b^{-4}}{b^{2}} = a^{2-(-1)} \cdot b^{-4-2} = a^{2+1} \cdot b^{-6} = a^{3} \cdot b^{-6} \]

### Step 3: Combine the results
Now, we combine the results from Step 1 and Step 2:
\[ (a^{-3} \cdot b^{4}) \times (a^{3} \cdot b^{-6}) \]
Using the property of exponents again:
\[ = a^{-3+3} \cdot b^{4-6} = a^{0} \cdot b^{-2} \]

### Step 4: Simplify further
Since \(a^{0} = 1\), we have:
\[ = 1 \cdot b^{-2} = \frac{1}{b^{2}} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{1}{b^{2}} \]