Mathematics 10 — Exercise Worked examples · Example 1 (iv)

Full question

Simplify the following expression.

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. First evaluate the positive power; a negative power is its reciprocal.

   $$\frac{1}{i^{25}}=\frac{1}{i^{25}}$$

5. Divide 25 by 4: 25 = 4 × 6 + 1.

   $$i^{25}=i^1\times {\left(i^4\right)}^6$$

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

   $$i^{25}=1^6\times i$$

7. Take the reciprocal of this result.

   $$\frac{1}{i^{25}}=\frac{1}{i}$$

8. Multiply numerator and denominator by i. Replace the denominator i² with −1, then simplify the signs.

   $$\frac{1}{i}=\frac{i}{ii}$$

9. The simplified value is therefore:

   $$\frac{1}{i^{25}}=-i$$

Answer

The final answer is shown below.