Mathematics 10 — Exercise 1.1 · Q1 (ii)

Full question

Simplify the following power of iota, using i² = −1.

Step-by-step solution

1. By definition, the square of iota is −1.

   $$i^2=-1$$

2. Multiply i² by i to obtain i³.

   $$i^3=-i$$

3. Square i² to obtain i⁴: (−1) × (−1) = 1.

   $$i^4={\left(-1\right)}^2$$

4. Divide 16 by 4: 16 = 4 × 4 + 0.

   $$i^{16}=i^0\times {\left(i^4\right)}^4$$

5. Every i⁴ contributes 1. Substitute the value of the remaining power.

   $$i^{16}=1^4\times 1$$

6. The simplified value is therefore:

   $$i^{16}=1$$

Answer

The final answer is shown below.