Compare :
`sqrt(24)` and `root (3)35`

Answer

To compare \( \sqrt{24} \) and \( \sqrt[3]{35} \), we will follow these steps:

### Step 1: Identify the roots
We have two numbers to compare:
1. \( \sqrt{24} \) (square root)
2. \( \sqrt[3]{35} \) (cube root)

### Step 2: Find a common root
To compare these two numbers, we can convert them to a common root. The least common multiple (LCM) of the indices (2 for square root and 3 for cube root) is 6. Therefore, we will convert both numbers to sixth roots.

### Step 3: Rewrite the first number
To convert \( \sqrt{24} \) to a sixth root, we can express it as follows:
\[ \sqrt{24} = 24^{1/2} = 24^{3/6} \]
This means we need to raise 24 to the power of 3 to keep the equality:
\[ \sqrt{24} = \sqrt[6]{24^3} \]

### Step 4: Rewrite the second number
Next, we convert \( \sqrt[3]{35} \) to a sixth root:
\[ \sqrt[3]{35} = 35^{1/3} = 35^{2/6} \]
This means we need to raise 35 to the power of 2 to keep the equality:
\[ \sqrt[3]{35} = \sqrt[6]{35^2} \]

### Step 5: Compare the two sixth roots
Now we have:
1. \( \sqrt{24} = \sqrt[6]{24^3} \)
2. \( \sqrt[3]{35} = \sqrt[6]{35^2} \)

To compare \( 24^3 \) and \( 35^2 \):
- Calculate \( 24^3 \):
\[ 24^3 = 24 \times 24 \times 24 = 13824 \]
- Calculate \( 35^2 \):
\[ 35^2 = 35 \times 35 = 1225 \]

### Step 6: Conclusion
Now we compare \( 13824 \) and \( 1225 \):
Since \( 13824 > 1225 \), we conclude that:
\[ \sqrt{24} > \sqrt[3]{35} \]

### Final Answer
Thus, \( \sqrt{24} \) is greater than \( \sqrt[3]{35} \).

---