Chapter Closure
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Exercise 184 Page 434
a Start by combining like terms.
c How can we factor a conjugate pair of binomials?
d Use the Properties of Exponents to simplify the expression.
e Recall that i^2=- 1.
f Start by calculating i^1, i^2, i^3, and i^4. Then calculate i^5, i^6, i^7, and i^8. Do you notice any patterns?
a 10+2i
b -16+30i
c 50
d -12i
e - i
f 1
a We can simplify the given expression by combining like terms. This involves treating the imaginary unit i as though it was a variable. In this case we need to remove the parentheses first.
b We can simplify the given expression by factoring a perfect square trinomial.
c We can simplify the given expression by factoring a conjugate pair of binomials.
d To simplify the given expression we will use the Properties of Exponents. Let's also recall the definition of the imaginary unit i.
(3i)(2i)^2
PowProdII
(a b)^m=a^m b^m
(3i)(2^2i^2)
▼
Simplify
SplitIntoFactors
Split into factors
3* i* 2^2* i^2
CalcPow
Calculate power
3* i* 4* i^2
IDefPow
i^2=- 1
3* i* 4* (-1)
CommutativePropMult
Commutative Property of Multiplication
(-1)* 3* 4 * i
Multiply
Multiply
-12i
e We can use the Properties of Exponents to simplify the given expression. To do so, let's also recall the definition of the imaginary unit i.
i^3
WriteSum
Write as a sum
i^(2+1)
▼
Simplify
SumInExponent
a^(m+n)=a^m*a^n
i^2* i^1
IDefPow
i^2=- 1
(-1)* i^1
ExponentOne
a^1=a
(-1)* i
MultNegPos
(- a)b = - ab
- i
f Generally, the value of i^n — where n is a whole number — can be calculated by dividing n by 4 and considering the remainder. Let R be the remainder when n is divided by 4.
Extra
Powers of iThe Commutative and Associative Properties of Multiplication hold true for imaginary numbers. We can use them to find the powers of i. Let's consider the first four powers. Recall that any number raised to the power of one equals itself, so i^1=i. Moreover, we know that i^2= - 1. ccccccc i^1&=& i i^2&=& - 1 i^3&=& i^2 * i &=& - 1 * i&=& - i i^4&=& i^2 * i^2&=& ( - 1) * ( - 1)&=& 1 Let's now calculate the following four powers. To do so, we will use the results obtained above. ccccccc i^5 &=& i^4 * i &=& 1 * i &=& i i^6 &=& i^4 * i^2 &=& 1 * ( - 1) &=& - 1 i^7 &=& i^4 * i^3 &=& 1 * ( - i) &=& - i i^8 &=& i^4 * i^4 &=& 1 * 1 &=& 1 Notice that the pattern i, - 1, - i, 1, ... repeats in that order continuously after the first four results.
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