Which one is smaller out of `root(3)(2) and root(4)(3)` ?

Answer

To determine which value is smaller between \( \sqrt[3]{2} \) (the cube root of 2) and \( \sqrt[4]{3} \) (the fourth root of 3), we can follow these steps:

### Step 1: Rewrite the expressions in exponential form
The cube root of 2 can be expressed as:
\[ \sqrt[3]{2} = 2^{1/3} \]
The fourth root of 3 can be expressed as:
\[ \sqrt[4]{3} = 3^{1/4} \]

### Step 2: Compare the exponents
To compare \( 2^{1/3} \) and \( 3^{1/4} \), we can rewrite them with a common base or compare their decimal values. However, it is often easier to compare their logarithmic values or calculate their numerical values directly.

### Step 3: Calculate the numerical values
Let's calculate the approximate numerical values of \( 2^{1/3} \) and \( 3^{1/4} \).

1. **Calculate \( 2^{1/3} \)**:
\[ 2^{1/3} \approx 1.2599 \]

2. **Calculate \( 3^{1/4} \)**:
\[ 3^{1/4} \approx 1.3161 \]

### Step 4: Compare the results
Now we can compare the two results:
- \( 2^{1/3} \approx 1.2599 \)
- \( 3^{1/4} \approx 1.3161 \)

Since \( 1.2599 < 1.3161 \), we conclude that:
\[ \sqrt[3]{2} < \sqrt[4]{3} \]

### Final Answer
Thus, \( \sqrt[3]{2} \) is smaller than \( \sqrt[4]{3} \).

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