If `r^(1//3) + (1)/(r^(1//3)) = 3` for a real number r not equal to 0 then what `r + (1)/(r )` equal to ?

Options

A. 27
B. 36
C. 9
D. 18

Answer

To solve the equation \( r^{1/3} + \frac{1}{r^{1/3}} = 3 \) and find the value of \( r + \frac{1}{r} \), we can follow these steps:

### Step 1: Cube both sides of the equation
Given:
\[ r^{1/3} + \frac{1}{r^{1/3}} = 3 \]
Cubing both sides:
\[ \left(r^{1/3} + \frac{1}{r^{1/3}}\right)^3 = 3^3 \]
This simplifies to:
\[ \left(r^{1/3} + \frac{1}{r^{1/3}}\right)^3 = 27 \]

### Step 2: Apply the formula for the cube of a sum
Using the identity \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \), where \( a = r^{1/3} \) and \( b = \frac{1}{r^{1/3}} \):
\[ a^3 = r, \quad b^3 = \frac{1}{r}, \quad ab = 1 \]
So we have:
\[ r + \frac{1}{r} + 3 \cdot 1 \cdot (r^{1/3} + \frac{1}{r^{1/3}}) = 27 \]

### Step 3: Substitute the known value
Substituting \( r^{1/3} + \frac{1}{r^{1/3}} = 3 \):
\[ r + \frac{1}{r} + 3 \cdot 3 = 27 \]
This simplifies to:
\[ r + \frac{1}{r} + 9 = 27 \]

### Step 4: Solve for \( r + \frac{1}{r} \)
Rearranging the equation gives:
\[ r + \frac{1}{r} = 27 - 9 \]
Thus:
\[ r + \frac{1}{r} = 18 \]

### Final Answer
Therefore, the value of \( r + \frac{1}{r} \) is \( 18 \).

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