The value of `(3+2sqrt2)^(-3)+(3-2sqrt2)^(-3)` is equal to

Options

A. 189
B. 180
C. 108
D. 198

Answer

To solve the expression \( (3 + 2\sqrt{2})^{-3} + (3 - 2\sqrt{2})^{-3} \), we will follow these steps:

### Step 1: Rewrite the expression with positive exponents
The negative exponent indicates that we can take the reciprocal:
\[ (3 + 2\sqrt{2})^{-3} = \frac{1}{(3 + 2\sqrt{2})^3} \]
\[ (3 - 2\sqrt{2})^{-3} = \frac{1}{(3 - 2\sqrt{2})^3} \]
Thus, we can rewrite the entire expression as:
\[ \frac{1}{(3 + 2\sqrt{2})^3} + \frac{1}{(3 - 2\sqrt{2})^3} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is:
\[ (3 + 2\sqrt{2})^3(3 - 2\sqrt{2})^3 \]
So we can write:
\[ \frac{(3 - 2\sqrt{2})^3 + (3 + 2\sqrt{2})^3}{(3 + 2\sqrt{2})^3(3 - 2\sqrt{2})^3} \]

### Step 3: Simplify the denominator
Using the difference of squares:
\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]
Thus, the denominator simplifies to:
\[ ((3 + 2\sqrt{2})(3 - 2\sqrt{2}))^3 = 1^3 = 1 \]

### Step 4: Simplify the numerator
Now we need to simplify the numerator:
\[ (3 - 2\sqrt{2})^3 + (3 + 2\sqrt{2})^3 \]
Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):
Let \( a = 3 + 2\sqrt{2} \) and \( b = 3 - 2\sqrt{2} \):
\[ a + b = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6 \]
\[ ab = (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 1 \]
Now calculate \( a^2 + b^2 \):
\[ a^2 + b^2 = (a + b)^2 - 2ab = 6^2 - 2 \cdot 1 = 36 - 2 = 34 \]
Thus:
\[ (3 - 2\sqrt{2})^3 + (3 + 2\sqrt{2})^3 = 6 \cdot (34 - 1) = 6 \cdot 33 = 198 \]

### Final Result
Since the denominator is 1, the final result is:
\[ (3 + 2\sqrt{2})^{-3} + (3 - 2\sqrt{2})^{-3} = 198 \]