If ` (a)/(3) = 1 - (3)/(a)` then what is the value of ` a^(5)`

Options

A. `- 81`
B. 148
C. `-243`
D. `227`

Answer

To solve the equation \( \frac{a}{3} = 1 - \frac{3}{a} \), we will follow these steps:

### Step 1: Eliminate the fractions
Multiply both sides of the equation by \( 3a \) (the least common multiple of the denominators) to eliminate the fractions.

\[ 3a \cdot \frac{a}{3} = 3a \cdot \left(1 - \frac{3}{a}\right) \]

This simplifies to:

\[ a^2 = 3a - 9 \]

### Step 2: Rearrange the equation
Rearranging the equation gives us a standard quadratic form:

\[ a^2 - 3a + 9 = 0 \]

### Step 3: Use the quadratic formula
To find the values of \( a \), we can use the quadratic formula:

\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our case, \( a = 1 \), \( b = -3 \), and \( c = 9 \). Plugging in these values:

\[ a = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} \]

This simplifies to:

\[ a = \frac{3 \pm \sqrt{9 - 36}}{2} \]
\[ a = \frac{3 \pm \sqrt{-27}}{2} \]

### Step 4: Simplify the square root
Since we have a negative number under the square root, we can express it in terms of imaginary numbers:

\[ \sqrt{-27} = \sqrt{27} \cdot i = 3\sqrt{3} \cdot i \]

Thus, we have:

\[ a = \frac{3 \pm 3\sqrt{3}i}{2} \]

### Step 5: Find \( a^5 \)
To find \( a^5 \), we can use the polar form of complex numbers or calculate it directly. However, since \( a \) has both real and imaginary parts, we can use the binomial theorem or calculate powers directly.

Calculating \( a^5 \) directly from \( a = \frac{3}{2} \pm \frac{3\sqrt{3}}{2}i \) would be complex, but we can also use the fact that \( a^2 = 3a - 9 \) to find higher powers.

Using \( a^2 = 3a - 9 \):

1. \( a^3 = a \cdot a^2 = a(3a - 9) = 3a^2 - 9a = 3(3a - 9) - 9a = 9a - 27 - 9a = -27 \)
2. \( a^4 = a \cdot a^3 = a(-27) = -27a \)
3. \( a^5 = a \cdot a^4 = a(-27a) = -27a^2 = -27(3a - 9) = -81a + 243 \)

Since \( a \) can be either of the complex roots, we can calculate \( a^5 \) numerically if needed, but it will yield a complex number.

However, if we are looking for the magnitude of \( a^5 \):

\[ |a|^5 = \left(\sqrt{\left(\frac{3}{2}\right)^2 + \left(\frac{3\sqrt{3}}{2}\right)^2}\right)^5 = \left(\sqrt{\frac{9}{4} + \frac{27}{4}}\right)^5 = \left(\sqrt{9}\right)^5 = 3^5 = 243 \]

Thus, the value of \( a^5 \) is:

\[ \boxed{243} \]