Exercise 67 Page 183

We are asked to simplify (1+2i)^3. Let's do this in two steps.

Simplifying (1+2i)^2

Let's simplify (1+2i)^2 first.

(1+2i)^2=(1+2i)(1+2i)

▼

Simplify right-hand side

Distr

Distribute (1+2i)

(1+2i)^2=(1+2i)+2i(1+2i)

Distr

Distribute 2i

(1+2i)^2=(1+2i)+(2i+4i^2)

RemovePar

Remove parentheses

(1+2i)^2=1+2i+2i+4i^2

IDefPow

i^2=- 1

(1+2i)^2=1+2i+2i+4(-1)

MultPosNeg

a(- b)=- a * b

(1+2i)^2=1+2i+2i+(-4)

AddNeg

a+(- b)=a-b

(1+2i)^2=1+2i+2i-4

AddSubTerms

Add and subtract terms

(1+2i)^2=-3+4i

Simplifying (1+2i)^3

To move forward, let's use the identity a^3=a^2a for a=(1+2i). After rewriting the expression, we can use our previous result and substitute (1+2i)^2 with -3+4i.

(1+2i)^3=(1+2i)^2(1+2i)

Substitute

(1+2i)^2= -3+4i

(1+2i)^3=( -3+4i)(1+2i)

▼

Simplify right-hand side

Distr

Distribute (1+2i)

(1+2i)^3=-3(1+2i)+4i(1+2i)

Distr

Distribute -3 & 4i

(1+2i)^3=-3-6i+4i+8i^2

IDefPow

i^2=- 1

(1+2i)^3=-3-6i+4i+8(-1)

MultPosNeg

a(- b)=- a * b

(1+2i)^3=-3-6i+4i+(-8)

AddNeg

a+(- b)=a-b

(1+2i)^3=-3-6i+4i-8

AddSubTerms

Add and subtract terms

(1+2i)^3=-11-2i

In two steps we found the simplification asked. (1+2i)^3=-11-2i

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Simplifying (1+2i)^2

Simplifying (1+2i)^3

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