1.1 Q-7

Exercise 1.1 — Question 7

Problem

Evaluate the following powers of $i$:

(i) $i^9$   (ii) $i^{14}$   (iii) $i^{-3}$   (iv) $i^{101}$

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Key Concept: Cyclic Nature of Powers of $i$

Since $i=\sqrt{-1}$, the powers of $i$ repeat in a cycle of 4:

$i^1=i,i^2=-1,i^3=-i,i^4=1$

$i^5=i,i^6=-1,\dots$

Rule: To evaluate $i^n$, divide $n$ by $4$ and use the remainder $r$:

$i^n=i^r,r=n\mathrm{mod}4$

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Solutions

(i) $i^9$

Divide $9$ by $4$: $9=4\times 2+1$, so remainder $=1$

$i^9=i^1=i$

(ii) $i^{14}$

Divide $14$ by $4$: $14=4\times 3+2$, so remainder $=2$

$i^{14}=i^2=-1$

(iii) $i^{-3}$

For negative powers, multiply numerator and denominator by a suitable power of $i$:

$i^{-3}=\frac{1}{i^3}=\frac{1}{-i}=\frac{1}{-i}\times \frac{i}{i}=\frac{i}{-i^2}=\frac{i}{-(-1)}=\frac{i}{1}=i$

Alternatively, $i^{-3}=i^{-3+4}=i^1=i$ (adding $4$ does not change the value).

(iv) $i^{101}$

Divide $101$ by $4$: $101=4\times 25+1$, so remainder $=1$

$i^{101}=i^1=i$

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Summary

The powers of $i$ cycle with period $4$. For any integer $n$:

$i^n=i^{n\mathrm{mod}4}$

For negative exponents, add a multiple of $4$ to make the exponent non-negative, or use $i^{-1}=-i$, $i^{-2}=-1$, $i^{-3}=i$, $i^{-4}=1$.