If a and b positive real numbers, which of the following is equivalent to `((2a^(-1)sqrt(b))^(4))/(ab^(-3))`?

Options

A. `8a^(2) b^(4)`
B. `(8 b ^(6))/(a^(4))`
C. `(16 b ^(5))/(a^(5))`
D. `(16 b ^(4))/(a^(5))`

Answer

To simplify the expression \(\frac{(2a^{-1}\sqrt{b})^{4}}{ab^{-3}}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{(2a^{-1}\sqrt{b})^{4}}{ab^{-3}} \]

### Step 2: Apply the power of a product
Using the property of exponents \((xy)^n = x^n y^n\), we can distribute the exponent \(4\) to each term inside the parentheses:
\[ = \frac{(2^{4})(a^{-1})^{4}(\sqrt{b})^{4}}{ab^{-3}} \]

### Step 3: Simplify each term
Now we simplify each term:
- \(2^{4} = 16\)
- \((a^{-1})^{4} = a^{-4}\)
- \((\sqrt{b})^{4} = (b^{1/2})^{4} = b^{2}\)

Substituting these back into the expression gives:
\[ = \frac{16 a^{-4} b^{2}}{ab^{-3}} \]

### Step 4: Rewrite the denominator
The denominator can be rewritten using the properties of exponents:
\[ ab^{-3} = a \cdot \frac{1}{b^{3}} = \frac{a}{b^{3}} \]

### Step 5: Invert and multiply
Now we can rewrite the expression as:
\[ = 16 a^{-4} b^{2} \cdot \frac{b^{3}}{a} \]

### Step 6: Combine the fractions
Now we combine the terms:
\[ = 16 \cdot \frac{b^{2} \cdot b^{3}}{a \cdot a^{4}} = 16 \cdot \frac{b^{5}}{a^{5}} \]

### Step 7: Final expression
This simplifies to:
\[ = \frac{16 b^{5}}{a^{5}} \]

### Conclusion
Thus, the expression \(\frac{(2a^{-1}\sqrt{b})^{4}}{ab^{-3}}\) simplifies to:
\[ \frac{16 b^{5}}{a^{5}} \]

### Final Answer
The equivalent expression is:
\[ \frac{16 b^{5}}{a^{5}} \]