Exercise 1.1 — Question 5

Question

Simplify the following powers of $i$:

(i) $i^7$    (ii) $i^{50}$    (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:

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

For any integer $n$, divide $n$ by $4$ and use the remainder $r$:

$i^n=i^r,r\in \{0,1,2,3\}$

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Solutions

(i) $i^7$

Divide $7$ by $4$: $7=4\times 1+3$, so remainder $=3$.

$i^7=i^3=-i$

(ii) $i^{50}$

Divide $50$ by $4$: $50=4\times 12+2$, so remainder $=2$.

$i^{50}=i^2=-1$

(iii) $i^{-3}$

For negative powers, use $i^{-1}=\frac{1}{i}=\frac{i}{i^2}=\frac{i}{-1}=-i$.

$i^{-3}=\frac{1}{i^3}=\frac{1}{-i}=\frac{1}{-i}\cdot \frac{i}{i}=\frac{i}{-i^2}=\frac{i}{1}=i$

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

To simplify $i^n$:

1. Find $r=n\mathrm{mod}4$ (remainder when $n$ is divided by $4$).
2. Use the table above to read off $i^r$.