If `a gt 1`, then arrange the following in ascending order.
I. `root3(root4(a^3))`
II. `root3(root5(a^4))`
III.`sqrt(root3(a))`
IV.`sqrt(root5(a^3))`

Options

A. I, II, III, IV
B. I, II, IV,III
C. IV, I, III, II
D. III, I, II, IV

Answer

To solve the problem of arranging the given expressions in ascending order, we will first simplify each expression step by step.

### Step 1: Simplify each expression

1. **Expression I: `root3(root4(a^3))`**
- This can be rewritten as:
\[ \text{Expression I} = \sqrt[3]{\sqrt[4]{a^3}} = a^{3/4 \cdot 1/3} = a^{1/4} \]

2. **Expression II: `root3(root5(a^4))`**
- This can be rewritten as:
\[ \text{Expression II} = \sqrt[3]{\sqrt[5]{a^4}} = a^{4/5 \cdot 1/3} = a^{4/15} \]

3. **Expression III: `sqrt(root3(a))`**
- This can be rewritten as:
\[ \text{Expression III} = \sqrt{\sqrt[3]{a}} = a^{1/3 \cdot 1/2} = a^{1/6} \]

4. **Expression IV: `sqrt(root5(a^3))`**
- This can be rewritten as:
\[ \text{Expression IV} = \sqrt{\sqrt[5]{a^3}} = a^{3/5 \cdot 1/2} = a^{3/10} \]

### Step 2: Compare the simplified expressions

Now we have the following simplified forms:
- I: \( a^{1/4} \)
- II: \( a^{4/15} \)
- III: \( a^{1/6} \)
- IV: \( a^{3/10} \)

### Step 3: Find a common denominator

To compare these fractions, we can find a common denominator. The least common multiple (LCM) of the denominators (4, 15, 6, and 10) is 60. We will convert each exponent to have a denominator of 60:

1. **For Expression I:**
\[ a^{1/4} = a^{15/60} \]

2. **For Expression II:**
\[ a^{4/15} = a^{16/60} \]

3. **For Expression III:**
\[ a^{1/6} = a^{10/60} \]

4. **For Expression IV:**
\[ a^{3/10} = a^{18/60} \]

### Step 4: Arrange in ascending order

Now we can compare the exponents:
- \( \frac{10}{60} \) (Expression III)
- \( \frac{15}{60} \) (Expression I)
- \( \frac{16}{60} \) (Expression II)
- \( \frac{18}{60} \) (Expression IV)

Thus, the ascending order of the expressions is:
1. Expression III: \( \sqrt{\sqrt[3]{a}} \) (or \( a^{1/6} \))
2. Expression I: \( \sqrt[3]{\sqrt[4]{a^3}} \) (or \( a^{1/4} \))
3. Expression II: \( \sqrt[3]{\sqrt[5]{a^4}} \) (or \( a^{4/15} \))
4. Expression IV: \( \sqrt{\sqrt[5]{a^3}} \) (or \( a^{3/10} \))

### Final Answer

The expressions in ascending order are:
**III, I, II, IV**