Suppose that `w = 2^(1//2), x = 3^(1//3),y = 6^(1//6)` and `z = 8^(1//8)`. From among these number list, the biggest, second biggest numbers are

Options

A. `w,x`
B. `x,w`
C. `y,z`
D. `x,z`

Answer

To find the biggest and second biggest numbers among \( w = 2^{1/2}, x = 3^{1/3}, y = 6^{1/6}, z = 8^{1/8} \), we will evaluate each expression step by step.

### Step 1: Calculate \( w \)
\[ w = 2^{1/2} = \sqrt{2} \approx 1.414 \]

**Hint:** Remember that \( 2^{1/2} \) is the square root of 2.

### Step 2: Calculate \( x \)
\[ x = 3^{1/3} \approx 1.442 \]

**Hint:** The cube root of 3 can be approximated using a calculator or known values.

### Step 3: Calculate \( y \)
\[ y = 6^{1/6} = (2 \cdot 3)^{1/6} = 2^{1/6} \cdot 3^{1/6} \]
Using logarithms or known values:
\[ y \approx 1.348 \]

**Hint:** You can break down \( 6^{1/6} \) into its prime factors for easier calculation.

### Step 4: Calculate \( z \)
\[ z = 8^{1/8} = (2^3)^{1/8} = 2^{3/8} \approx 1.090 \]

**Hint:** Remember that \( 8 = 2^3 \) and use the power of a power rule.

### Step 5: Compare the values
Now we have the approximate values:
- \( w \approx 1.414 \)
- \( x \approx 1.442 \)
- \( y \approx 1.348 \)
- \( z \approx 1.090 \)

### Step 6: Identify the biggest and second biggest
From the calculated values:
- The biggest number is \( x \approx 1.442 \).
- The second biggest number is \( w \approx 1.414 \).

### Conclusion
Thus, the biggest number is \( x \) and the second biggest number is \( w \).

**Final Answer:**
- Biggest: \( x \)
- Second Biggest: \( w \)