PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
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Roots
The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form o…
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Chapter P.6, Problem 51E
Solution

(a)

To simplify the complex numbers i,i2,i3....,i8 by evaluating each one.

  −1,−1,−i,1.....1

Given:

The complex number, i,i2,i3....,i8 .

Concept used:

  • Properties of complex number: −1=i, i2=−1, i3=−i

Calculation:

In order to simplify the complex number ,solving the complex number one by one by using the properties of complex number as shown below:

  i=−1

  i2=i×ii2=−1×−1i2=−1

  i3=i2×ii3=−1×ii3=−ii4=i2×i2i4=−1×−1i4=1i5=i3×i2i5=−i×−1i5=ii6=i4×i2i6=1×−1i6=−1i7=i4×i3i7=1×−ii7=−ii8=i4×i4i8=1×1i8=1

So, the complex number after simplification are −1,−1,−i,1.....1

(b)

To simplify the complex numbers i−1,i−2,i−3....,i−8 by evaluating each one.

  −i,−1,i,1....,1

Given:

The complex number, i−1,i−2,i−3....,i−8 .

Concept used:

  • Properties of complex number: −1=i, i2=−1, i3=−i

Calculation:

In order to simplify the complex number ,first write the complex number in the inverse format then solving the complex number one by one by using the properties of complex number as shown below:

  i−1=1ii−1=1i×−i−ii−1=−ii−2=1i2i−2=1−1i−2=−1i−3=1i3i−3=1−ii−3=1−i×iii−3=ii−4=1i4i−4=1i4i−4=1i−5=1i5i−5=i8i5i−5=i3i−5=−ii−6=1i6i−6=i8i6i−6=i2i−6=−1i−7=1i7i−7=i8i7i−7=ii−8=1i8i−8=i8i8i−8=11i−8=1

So, the complex number after simplification are −i,−1,i,1....,1

(c)

To simplify the complex numbers i0 .

  i0=1

Given:

The complex number, i0 .

Concept used:

  • Properties of complex number: −1=i, i2=−1, i3=−i

Calculation:

In order to simplify the complex number ,by using the properties of complex number as shown below:

  i0=1

so, the value of i0=1

(d)

To discuss your result from a to c with the member of your group and write a summary about the integral power of i.

As i discuss the result with by group members about the integral power of i , as we all know that the value of I i0=1, i2=−1, i3=−i, i4=1

We have to remember these values to simplify any complex numbers.

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Chapter P.6, Problem 50E
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Chapter P.6, Problem 52E

Chapter P Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

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