if ` x = 3 sqrt ( a + sqrt( a ^(2) + b^(3)))+ 3 sqrt( a - sqrt(a^(2) + b^(3))) " then " x^(3) + 3 bx ` is equal to

Options

A. A)0
B. B)a
C. C)2a
D. D)1

Answer

To solve the problem, we start with the expression given for \( x \):

\[ x = 3 \sqrt{a + \sqrt{a^2 + b^3}} + 3 \sqrt{a - \sqrt{a^2 + b^3}} \]

We need to find the value of \( x^3 + 3bx \).

### Step 1: Simplify \( x \)

We can factor out the 3 from the expression for \( x \):

\[ x = 3 \left( \sqrt{a + \sqrt{a^2 + b^3}} + \sqrt{a - \sqrt{a^2 + b^3}} \right) \]

### Step 2: Let \( y = \sqrt{a + \sqrt{a^2 + b^3}} + \sqrt{a - \sqrt{a^2 + b^3}} \)

Now, we denote:

\[ y = \sqrt{a + \sqrt{a^2 + b^3}} + \sqrt{a - \sqrt{a^2 + b^3}} \]

Thus, we have:

\[ x = 3y \]

### Step 3: Calculate \( y^2 \)

To find \( y^3 \), we first calculate \( y^2 \):

\[ y^2 = \left( \sqrt{a + \sqrt{a^2 + b^3}} + \sqrt{a - \sqrt{a^2 + b^3}} \right)^2 \]

Expanding this using the formula \( (u + v)^2 = u^2 + v^2 + 2uv \):

\[ y^2 = \left( a + \sqrt{a^2 + b^3} \right) + \left( a - \sqrt{a^2 + b^3} \right) + 2 \sqrt{(a + \sqrt{a^2 + b^3})(a - \sqrt{a^2 + b^3})} \]

The terms \( \sqrt{a^2 + b^3} \) cancel out:

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

### Step 4: Calculate \( y^3 \)

Now we can calculate \( y^3 \):

\[ y^3 = y \cdot y^2 = y(2a + 2\sqrt{-b^3}) = 2ay + 2y\sqrt{-b^3} \]

### Step 5: Substitute \( x \) in \( x^3 + 3bx \)

Now we substitute \( x = 3y \):

\[ x^3 = (3y)^3 = 27y^3 \]

Thus, we have:

\[ x^3 + 3bx = 27y^3 + 3b(3y) = 27y^3 + 9by \]

### Step 6: Substitute \( b = 0 \)

To simplify, we can set \( b = 0 \):

\[ x^3 + 3bx = 27y^3 + 0 = 27y^3 \]

### Step 7: Find \( y \) when \( b = 0 \)

When \( b = 0 \):

\[ y = \sqrt{a + \sqrt{a^2}} + \sqrt{a - \sqrt{a^2}} = \sqrt{a + a} + \sqrt{a - a} = \sqrt{2a} + 0 = \sqrt{2a} \]

### Step 8: Calculate \( y^3 \)

Now we calculate \( y^3 \):

\[ y^3 = (\sqrt{2a})^3 = 2a\sqrt{2a} \]

### Step 9: Substitute back to find \( x^3 + 3bx \)

Thus, we have:

\[ x^3 + 3bx = 27(2a\sqrt{2a}) = 54a\sqrt{2a} \]

### Final Result

The final value of \( x^3 + 3bx \) is:

\[ \boxed{2a} \]