Which one of the following sets of surds is in correct sequence of ascending order of their values?

Options

A. `root(4)(10), root(3)(6), sqrt(3)`
B. `sqrt(3), root(4)(10), root(3)(6)`
C. `sqrt(3), root(3)(6), root(4)(10)`
D. `root(4)(10), sqrt(3), root(3)(6) `

Answer

To determine which set of surds is in the correct sequence of ascending order, we will follow these steps:

### Step 1: Identify the Surds
We have three surds to compare:
1. \( A = 10^{1/4} \)
2. \( B = 6^{1/3} \)
3. \( C = 3^{1/2} \)

### Step 2: Rewrite the Surds with Common Denominator
To compare these surds, we need to express them with a common denominator in their exponents. The denominators are 4, 3, and 2. The least common multiple (LCM) of these numbers is 12.

We will rewrite each surd:
- For \( A = 10^{1/4} \), we multiply the exponent by \( \frac{3}{3} \):
\[ A = 10^{1/4} = 10^{3/12} \]

- For \( B = 6^{1/3} \), we multiply the exponent by \( \frac{4}{4} \):
\[ B = 6^{1/3} = 6^{4/12} \]

- For \( C = 3^{1/2} \), we multiply the exponent by \( \frac{6}{6} \):
\[ C = 3^{1/2} = 3^{6/12} \]

### Step 3: Compare the Values
Now we can compare:
- \( A = 10^{3/12} \)
- \( B = 6^{4/12} \)
- \( C = 3^{6/12} \)

### Step 4: Calculate the Values
Next, we will calculate the values of \( 10^{3/12} \), \( 6^{4/12} \), and \( 3^{6/12} \).

1. **Calculate \( 10^{3/12} \)**:
\[ 10^{3/12} = 10^{0.25} \approx 1.778 \]

2. **Calculate \( 6^{4/12} \)**:
\[ 6^{4/12} = 6^{0.333} \approx 1.817 \]

3. **Calculate \( 3^{6/12} \)**:
\[ 3^{6/12} = 3^{0.5} = \sqrt{3} \approx 1.732 \]

### Step 5: Order the Values
Now we can order the values:
- \( C \approx 1.732 \) (smallest)
- \( A \approx 1.778 \)
- \( B \approx 1.817 \) (largest)

Thus, the correct ascending order is:
\[ C < A < B \]

### Conclusion
The correct sequence in ascending order of their values is \( 3^{1/2} < 10^{1/4} < 6^{1/3} \).