If ` a = sqrt( 7 + 2 sqrt( 12)) and b = sqrt( 7 - 2 sqrt( 12) ) " then " ( a^(3) + b^(3))` is equal to

Options

A. 40
B. 44
C. 48
D. 52

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 \) where \( a = \sqrt{7 + 2\sqrt{12}} \) and \( b = \sqrt{7 - 2\sqrt{12}} \).

### Step 1: Simplifying \( a \) and \( b \)

First, we simplify \( a \):

\[ a = \sqrt{7 + 2\sqrt{12}} \]

We can rewrite \( \sqrt{12} \) as \( 2\sqrt{3} \):

\[ a = \sqrt{7 + 2 \cdot 2\sqrt{3}} = \sqrt{7 + 4\sqrt{3}} \]

Next, we can express \( 7 + 4\sqrt{3} \) in a different form. We can try to express it as a square:

\[ 7 + 4\sqrt{3} = (2 + \sqrt{3})^2 \]

Calculating \( (2 + \sqrt{3})^2 \):

\[ (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

Thus, we have:

\[ a = 2 + \sqrt{3} \]

Now, we simplify \( b \):

\[ b = \sqrt{7 - 2\sqrt{12}} = \sqrt{7 - 4\sqrt{3}} \]

Similarly, we can express \( 7 - 4\sqrt{3} \) as a square:

\[ 7 - 4\sqrt{3} = (2 - \sqrt{3})^2 \]

Calculating \( (2 - \sqrt{3})^2 \):

\[ (2 - \sqrt{3})^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

Thus, we have:

\[ b = 2 - \sqrt{3} \]

### Step 2: Finding \( a + b \) and \( ab \)

Now, we calculate \( a + b \):

\[ a + b = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 \]

Next, we calculate \( ab \):

\[ ab = (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 3: Using the identity for \( a^3 + b^3 \)

We can use the identity:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

We already have \( a + b = 4 \) and \( ab = 1 \). Now we need \( a^2 + b^2 \):

\[ a^2 + b^2 = (a + b)^2 - 2ab = 4^2 - 2 \cdot 1 = 16 - 2 = 14 \]

Thus, we can find \( a^2 - ab + b^2 \):

\[ a^2 - ab + b^2 = a^2 + b^2 - ab = 14 - 1 = 13 \]

### Step 4: Final calculation for \( a^3 + b^3 \)

Now we can substitute back into the identity:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) = 4 \cdot 13 = 52 \]

### Final Answer

Thus, the value of \( a^3 + b^3 \) is:

\[ \boxed{52} \]