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Core Connections Integrated III, 2015

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Core Connections Integrated III, 2015 View details

1. Section 7.1

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Exercise 40 Page 332

Knowing that $i^{1} = i$ and by the definition of the imaginary unit $i^{2} = - 1,$ how can you obtain the next powers of $i ?$

See solution.

Practice makes perfect

To find the powers of

i,

keep in mind that the Commutative and Associative Properties of Multiplication hold true for imaginary numbers. This allows us to rewrite the powers as products.

i^{3} i^{4} = = i^{2} \cdot i i^{2} \cdot i^{2}

Now, recall that any number raised to the power of one equals itself, so

i^{1} = i .

Moreover — by definition the definition of the imaginary unit — we know that

i^{2} = - 1 .

i^{1} i^{2} i^{3} i^{4} = = = = i - 1 i^{2} \cdot i i^{2} \cdot i^{2} = = - 1 \cdot i (- 1) \cdot (- 1) = = - i 1

Therefore,

i^{3} = - i

and

i^{4} = 1 .

Subchapter links

7.1.1 Using Logarithms to Solve Exponential Equations p.323

7.1.2 Investigating the Properties of Logarithms p.327-328

7.1.3 Writing Equations of Exponential Functions p.331-332

7.1.4 An Application of Logarithms p.336-337