`root(3)(1+sqrt(2)).root(6)(3-2sqrt2)=`?

Options

A. None of these
B. `sqrt(2)-1`
C. 1
D. `3-2sqrt(2)`

Answer

To solve the expression \( \sqrt[3]{1+\sqrt{2}} \cdot \sqrt[6]{3-2\sqrt{2}} \), we will follow these steps:

### Step 1: Simplify \( \sqrt[6]{3 - 2\sqrt{2}} \)

We start with the expression \( 3 - 2\sqrt{2} \). We can rewrite this expression in a different form. Notice that:

\[ 3 - 2\sqrt{2} = (\sqrt{2} - 1)^2 \]

This is because:

\[ (\sqrt{2} - 1)^2 = (\sqrt{2})^2 - 2 \cdot \sqrt{2} \cdot 1 + 1^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]

Thus, we can rewrite \( \sqrt[6]{3 - 2\sqrt{2}} \) as:

\[ \sqrt[6]{(\sqrt{2} - 1)^2} \]

### Step 2: Apply the exponent rule

Using the property of exponents, we have:

\[ \sqrt[6]{(\sqrt{2} - 1)^2} = (\sqrt{2} - 1)^{\frac{2}{6}} = (\sqrt{2} - 1)^{\frac{1}{3}} \]

### Step 3: Combine with \( \sqrt[3]{1 + \sqrt{2}} \)

Now we can combine this with \( \sqrt[3]{1 + \sqrt{2}} \):

\[ \sqrt[3]{1 + \sqrt{2}} \cdot (\sqrt{2} - 1)^{\frac{1}{3}} \]

### Step 4: Use the property of cube roots

Using the property of cube roots, we can combine these:

\[ \sqrt[3]{(1 + \sqrt{2}) \cdot (\sqrt{2} - 1)} \]

### Step 5: Simplify the expression inside the cube root

Now, we simplify \( (1 + \sqrt{2})(\sqrt{2} - 1) \):

\[ (1 + \sqrt{2})(\sqrt{2} - 1) = 1\cdot\sqrt{2} - 1 + \sqrt{2}\cdot\sqrt{2} - \sqrt{2} = 2 - 1 = 1 \]

### Step 6: Final simplification

Thus, we have:

\[ \sqrt[3]{1} = 1 \]

### Conclusion

The final answer is:

\[ \sqrt[3]{1+\sqrt{2}} \cdot \sqrt[6]{3-2\sqrt{2}} = 1 \]