Which of the following equivalent to the expressions `((a^((1)/(4))b^(3))/(a^(2)b^((1)/(2)))`?
Where `agt1 and bgt1`

Options

A. `(b^(3)sqrt(b))/(a^(4)sqrt(a^(3)))`
B. `(b^(3)sqrt(b))/(a^(3)sqrt(a))`
C. `(b^(3)sqrt(b))/(root(r)(a^(2)))`
D. `(sqrt(b^(3))/(root(4)(a^(2))`.

Answer

To solve the expression \(\frac{a^{\frac{1}{4}} b^{3}}{a^{2} b^{\frac{1}{2}}}\), we will simplify it step by step.

### Step 1: Rewrite the expression
The given expression is:
\[ \frac{a^{\frac{1}{4}} b^{3}}{a^{2} b^{\frac{1}{2}}} \]

### Step 2: Apply the laws of exponents
Using the property of exponents \(\frac{x^m}{x^n} = x^{m-n}\), we can simplify the expression for both \(a\) and \(b\).

For the \(a\) terms:
\[ \frac{a^{\frac{1}{4}}}{a^{2}} = a^{\frac{1}{4} - 2} = a^{\frac{1}{4} - \frac{8}{4}} = a^{-\frac{7}{4}} \]

For the \(b\) terms:
\[ \frac{b^{3}}{b^{\frac{1}{2}}} = b^{3 - \frac{1}{2}} = b^{\frac{6}{2} - \frac{1}{2}} = b^{\frac{5}{2}} \]

### Step 3: Combine the results
Now we can combine the results from the \(a\) and \(b\) simplifications:
\[ \frac{a^{\frac{1}{4}} b^{3}}{a^{2} b^{\frac{1}{2}}} = a^{-\frac{7}{4}} b^{\frac{5}{2}} \]

### Step 4: Rewrite with positive exponents
To express \(a^{-\frac{7}{4}}\) with a positive exponent, we can write:
\[ a^{-\frac{7}{4}} = \frac{1}{a^{\frac{7}{4}}} \]
Thus, the expression becomes:
\[ \frac{b^{\frac{5}{2}}}{a^{\frac{7}{4}}} \]

### Step 5: Final expression
The final simplified expression is:
\[ \frac{b^{\frac{5}{2}}}{a^{\frac{7}{4}}} \]