Mathematics 10 — Exercise 1.1 · Q1 (vii)

Full question

Simplify the following expression.

Step-by-step solution

1. For the same base, subtract denominator exponents: i⁸/i⁵ = i³.

   $$\frac{i^8}{i^5}=i^3$$

2. Raise a power to a power by multiplying exponents: 3 × (−5) = −15.

   $$\frac{1}{{\left(i^3\right)}^5}=\frac{1}{i^{15}}$$

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

   $$i^2=-1$$

4. Multiply i² by i to obtain i³.

   $$i^3=-i$$

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

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

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

   $$\frac{1}{i^{15}}=\frac{1}{i^{15}}$$

7. Divide 15 by 4: 15 = 4 × 3 + 3.

   $$i^{15}=i^3\times {\left(i^4\right)}^3$$

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

   $$i^{15}=1^3\times \left(-i\right)$$

9. Take the reciprocal of this result.

   $$\frac{1}{i^{15}}=\frac{1}{-i}$$

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

    $$\frac{1}{-i}=\frac{i}{\left(-i\right)\times i}$$

11. The simplified value is therefore:

    $$\frac{1}{i^{15}}=i$$

Answer

The final answer is shown below.