Write the following in ascending order of magniude .
`root(6)(6), root(3)(7),root(4)(8)`

Answer

To solve the problem of arranging the numbers \( \sqrt[6]{6}, \sqrt[3]{7}, \sqrt[4]{8} \) in ascending order of magnitude, we will follow these steps:

### Step 1: Rewrite the roots in exponential form
We can express the roots in terms of powers:
- \( \sqrt[6]{6} = 6^{1/6} \)
- \( \sqrt[3]{7} = 7^{1/3} \)
- \( \sqrt[4]{8} = 8^{1/4} \)

### Step 2: Find a common exponent
To compare these numbers, we need to express them with a common exponent. The least common multiple (LCM) of the denominators (6, 3, and 4) is 12. We will rewrite each expression with an exponent of \( \frac{1}{12} \).

### Step 3: Rewrite each expression with the common exponent
- For \( 6^{1/6} \):
\[ 6^{1/6} = 6^{2/12} = (6^2)^{1/12} = 36^{1/12} \]

- For \( 7^{1/3} \):
\[ 7^{1/3} = 7^{4/12} = (7^4)^{1/12} = 2401^{1/12} \]

- For \( 8^{1/4} \):
\[ 8^{1/4} = 8^{3/12} = (8^3)^{1/12} = 512^{1/12} \]

### Step 4: Compare the bases
Now we need to compare the bases:
- \( 36 \)
- \( 2401 \)
- \( 512 \)

### Step 5: Determine the order
Now we can compare the numbers:
- \( 36 < 512 < 2401 \)

### Step 6: Write the final answer in ascending order
Thus, the order from smallest to largest is:
\[ \sqrt[6]{6} < \sqrt[4]{8} < \sqrt[3]{7} \]

### Final Answer
The ascending order of the given roots is:
\[ \sqrt[6]{6}, \sqrt[4]{8}, \sqrt[3]{7} \]

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