Mathematics 10 — Exercise 1.1 · Q1 (iv)

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

   $$\frac{1}{i^{26}}=\frac{1}{i^{26}}$$

5. Divide 26 by 4: 26 = 4 × 6 + 2.

   $$i^{26}=i^2\times {\left(i^4\right)}^6$$

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

   $$i^{26}=1^6\times \left(-1\right)$$

7. Take the reciprocal of this result.

   $$\frac{1}{i^{26}}=\frac{1}{-1}$$

8. The simplified value is therefore:

   $$\frac{1}{i^{26}}=-1$$

9. Now multiply by the outside coefficient 27.

   $$\frac{27}{i^{26}}=-27$$

Answer

The final answer is shown below.