`root3(20+14sqrt(2))+root3(20-14sqrt(2))=4`. (a) true (b) false

Answer

To solve the equation \( \sqrt[3]{20 + 14\sqrt{2}} + \sqrt[3]{20 - 14\sqrt{2}} = 4 \), we will follow these steps:

### Step 1: Let \( a = \sqrt[3]{20 + 14\sqrt{2}} \) and \( b = \sqrt[3]{20 - 14\sqrt{2}} \)
We can rewrite the equation as:
\[ a + b = 4 \]

### Step 2: Calculate \( a^3 + b^3 \)
Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \), we first find \( a^3 \) and \( b^3 \):
\[ a^3 = 20 + 14\sqrt{2} \]
\[ b^3 = 20 - 14\sqrt{2} \]
Now, adding these:
\[ a^3 + b^3 = (20 + 14\sqrt{2}) + (20 - 14\sqrt{2}) = 40 \]

### Step 3: Calculate \( ab \)
Next, we find \( ab \):
\[ ab = \sqrt[3]{(20 + 14\sqrt{2})(20 - 14\sqrt{2})} \]
Using the difference of squares:
\[ (20 + 14\sqrt{2})(20 - 14\sqrt{2}) = 20^2 - (14\sqrt{2})^2 = 400 - 392 = 8 \]
Thus,
\[ ab = \sqrt[3]{8} = 2 \]

### Step 4: Use the identity for \( a^3 + b^3 \)
Now we can use the identity:
\[ a^3 + b^3 = (a + b)((a + b)^2 - 3ab) \]
Substituting the known values:
\[ 40 = 4(4^2 - 3 \cdot 2) \]
Calculating inside the parentheses:
\[ 40 = 4(16 - 6) = 4 \cdot 10 = 40 \]

### Conclusion
Since both sides of the equation are equal, we conclude that the original statement is true:
\[ \sqrt[3]{20 + 14\sqrt{2}} + \sqrt[3]{20 - 14\sqrt{2}} = 4 \text{ is true.} \]

### Final Answer
(a) true