If a is positive and `a^2 + 1/(a^2)` = 7, then `a^3 + 1/(a^3)` =

Options

A. 21
B. `3sqrt7`
C. 18
D. `7sqrt7`

Answer

To solve the problem where \( a^2 + \frac{1}{a^2} = 7 \) and we need to find \( a^3 + \frac{1}{a^3} \), we can follow these steps:

### Step 1: Use the identity for \( a^2 + \frac{1}{a^2} \)

We know that:
\[ a^2 + \frac{1}{a^2} = \left(a + \frac{1}{a}\right)^2 - 2 \]
Let \( x = a + \frac{1}{a} \). Then we can rewrite the equation as:
\[ x^2 - 2 = 7 \]

### Step 2: Solve for \( x \)

Rearranging the equation gives:
\[ x^2 = 7 + 2 = 9 \]
Taking the square root of both sides, we find:
\[ x = 3 \quad (\text{since } a \text{ is positive}) \]

### Step 3: Use the identity for \( a^3 + \frac{1}{a^3} \)

Now, we can use the identity:
\[ a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right)^3 - 3\left(a + \frac{1}{a}\right) \]
Substituting \( x = 3 \) into the equation gives:
\[ a^3 + \frac{1}{a^3} = 3^3 - 3 \cdot 3 \]

### Step 4: Calculate \( a^3 + \frac{1}{a^3} \)

Calculating the right-hand side:
\[ 3^3 = 27 \]
\[ 3 \cdot 3 = 9 \]
Thus,
\[ a^3 + \frac{1}{a^3} = 27 - 9 = 18 \]

### Final Answer

Therefore, the value of \( a^3 + \frac{1}{a^3} \) is \( \boxed{18} \).

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