Write in ascending order:
`2 root(3)5` and `3root(3)2`

Answer

To solve the problem of writing \(2 \sqrt[3]{5}\) and \(3 \sqrt[3]{2}\) in ascending order, we will follow these steps:

### Step 1: Rewrite the Numbers
We start by rewriting the two numbers clearly:
- First number: \(2 \sqrt[3]{5}\)
- Second number: \(3 \sqrt[3]{2}\)

### Step 2: Express in Terms of Cube Roots
We can express these numbers in terms of their cube roots:
- \(2 \sqrt[3]{5} = 2 \cdot 5^{1/3}\)
- \(3 \sqrt[3]{2} = 3 \cdot 2^{1/3}\)

### Step 3: Cube Both Sides to Eliminate the Cube Roots
To compare the two numbers, we can cube both expressions. This will help us eliminate the cube roots:
- For the first number:
\[ (2 \sqrt[3]{5})^3 = 2^3 \cdot 5 = 8 \cdot 5 = 40 \]
- For the second number:
\[ (3 \sqrt[3]{2})^3 = 3^3 \cdot 2 = 27 \cdot 2 = 54 \]

### Step 4: Compare the Results
Now we compare the results of the cubes:
- First number cubed: \(40\)
- Second number cubed: \(54\)

Since \(40 < 54\), we can conclude that:
\[ 2 \sqrt[3]{5} < 3 \sqrt[3]{2} \]

### Step 5: Write in Ascending Order
Based on our comparison, we can write the numbers in ascending order:
\[ 2 \sqrt[3]{5}, 3 \sqrt[3]{2} \]

### Final Answer
Thus, the final answer in ascending order is:
\[ 2 \sqrt[3]{5} < 3 \sqrt[3]{2} \]

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