Question

If `a = 1/(4 - a)`, find the value of `a^3 + 1/a^3`.

Answer 1

Given: `a = 1/(4 - a)`

Step-wise calculation:

1. Multiply both sides by (4 – a):

a(4 – a) = 1

2. Expand the left side:

4a – a² = 1

3. Rearrange into a quadratic equation:

a² – 4a + 1 = 0

4. From the quadratic equation, find `a + 1/a`:

Dividing the equation a² – 4a + 1 = 0 by (a) assuming (a ≠ 0):

`a - 4 + 1/a = 0`

⇒ `a + 1/a = 4`

5. Use the identity to find `a^3 + 1/a^3`:

`(a + 1/a)^3 = a^3 + 1/a^3 + 3(a + 1/a)`

6. Substitute `a + 1/a = 4`:

`4^3 = a^3 + 1/a^3 + 3 xx 4`

`64 = a^3 + 1/a^3 + 12`

7. Solve for `a^3 + 1/a^3`:

`a^3 + 1/a^3 = 64 - 12`

`a^3 + 1/a^3 = 52`