BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter I, Problem 38E
To determine

To find: The roots of the fifths root of 32 and sketch the roots in the complex plane.

Expert Solution

Answer to Problem 38E

The roots of fifth root of 32 are wk=118[cos(0+2kπ8)+isin(0+2kπ8)]=coskπ8+isinkπ8  where k=0,1,2,,7.

Explanation of Solution

Theorem used:

Roots of a complex number:

Let z=r(cosθ+isinθ) and n be a positive integer. Then z has the n distinct nth roots wk=r1n[cos(θ+2kπn)+isin(θ+2kπn)] where k=0,1,2,,n1.

Calculation:

Rewrite the complex number 32 in polar form.

The polar form of the complex number z=a+bi is z=r(cosθ+isinθ) where r=|z|=a2+b2  and tanθ=ba.

Consider the complex number 32.

Obtain the argument of the complex number 32.

tanθ=032=0

Thus, the argument of the complex number 32 is θ=tan1(0)=0

Obtain the modulus of the complex number 32.

r=|32|=322+(0)2=322=32

Thus, the value of r=32.

Therefore, the polar form of the complex number 32 is 32=32(cos0+isin0).

By the above theorem, the roots of eighth root of 32 are wk=118[cos(0+2kπ8)+isin(0+2kπ8)]=coskπ8+isinkπ8 where k=0,1,2,,5.

Use online calculator to sketch the roots in the complex plane as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter I, Problem 38E

From figure 1, it is observed that all fifths roots of 32 form a polygon on complex plane.

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