For what value of k, the expression `x^(6) - 18 x^(3) + k ` will be a perfect square ?

Options

A. `-9`
B. `- 81`
C. `+ 9`
D. `+ 81`

Answer

To find the value of \( k \) such that the expression \( x^6 - 18x^3 + k \) is a perfect square, we can follow these steps:

### Step 1: Rewrite the expression
We can rewrite the expression as:
\[ x^6 - 18x^3 + k \]
Notice that \( x^6 \) can be expressed as \( (x^3)^2 \). Thus, we can rewrite the expression as:
\[ (x^3)^2 - 18x^3 + k \]

### Step 2: Identify the form of a perfect square
A perfect square trinomial can be expressed in the form:
\[ A^2 - 2AB + B^2 = (A - B)^2 \]
In our case, we can let \( A = x^3 \) and \( B = \sqrt{k} \). Therefore, we want to express our expression in this form.

### Step 3: Set up the equation
From the perfect square form, we can identify:
\[ -2AB = -18x^3 \]
This implies:
\[ 2 \cdot x^3 \cdot \sqrt{k} = 18x^3 \]

### Step 4: Solve for \( \sqrt{k} \)
Dividing both sides by \( x^3 \) (assuming \( x \neq 0 \)):
\[ 2\sqrt{k} = 18 \]
Now, divide both sides by 2:
\[ \sqrt{k} = 9 \]

### Step 5: Solve for \( k \)
Now, squaring both sides gives:
\[ k = 9^2 = 81 \]

### Conclusion
Thus, the value of \( k \) for which the expression \( x^6 - 18x^3 + k \) is a perfect square is:
\[ \boxed{81} \]