For all positive values of g and h, which of the following expressions is equivalent to `g^(2) sqrt(g^5) cdot h^(2) root(4)(h^5)?`

Options

A. `g^(2) h^(2) root(5)(g^2h^2)`
B. `g^(3) h root(4)(g^2h^3)`
C. `g^(4) h^(3) root(4)(g^2h)`
D. `g^(4) h^(4) root(4)(g^2h)`

Answer

To solve the expression \( g^2 \sqrt{g^5} \cdot h^2 \sqrt[4]{h^5} \), we will simplify it step by step.

### Step 1: Rewrite the square root and fourth root
We start by rewriting the square root and fourth root in terms of exponents:
\[ \sqrt{g^5} = g^{5/2} \quad \text{and} \quad \sqrt[4]{h^5} = h^{5/4} \]

### Step 2: Substitute back into the expression
Now we substitute these back into the original expression:
\[ g^2 \cdot g^{5/2} \cdot h^2 \cdot h^{5/4} \]

### Step 3: Combine the exponents for \( g \)
For the \( g \) terms, we can combine the exponents:
\[ g^2 \cdot g^{5/2} = g^{2 + 5/2} = g^{4/2 + 5/2} = g^{9/2} \]

### Step 4: Combine the exponents for \( h \)
For the \( h \) terms, we also combine the exponents:
\[ h^2 \cdot h^{5/4} = h^{2 + 5/4} = h^{8/4 + 5/4} = h^{13/4} \]

### Step 5: Write the final expression
Now we can write the combined expression:
\[ g^{9/2} \cdot h^{13/4} \]

### Step 6: Rewrite \( g^{9/2} \)
We can rewrite \( g^{9/2} \) in terms of square roots:
\[ g^{9/2} = g^4 \cdot g^{1/2} = g^4 \sqrt{g} \]

### Step 7: Final expression
Thus, the final expression becomes:
\[ g^4 \sqrt{g} \cdot h^{13/4} \]

### Step 8: Rewrite \( h^{13/4} \)
We can also rewrite \( h^{13/4} \):
\[ h^{13/4} = h^3 \cdot h^{1/4} \]

### Final Result
Putting it all together, we have:
\[ g^4 \sqrt{g} \cdot h^3 \cdot \sqrt[4]{h} \]

### Conclusion
The expression \( g^2 \sqrt{g^5} \cdot h^2 \sqrt[4]{h^5} \) simplifies to:
\[ g^4 h^3 \sqrt{g} \sqrt[4]{h} \]