The value of ( 3 + i ) 7 in standard form by using De Moivre’s Theorem.

Question

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Answer 1

Question

Chapter 7.3, Problem 3E

(a)

To determine

The value of (3+i)7 in standard form by using De Moivre’s Theorem.

(b)

To determine

The value of (32+12i)21 in standard form by using De Moivre’s Theorem.

(c)

To determine

The value of (− 3+i)10 in standard form by using De Moivre’s Theorem.

(d)

To determine

The value of ( 32−12i)18 in standard form by using De Moivre’s Theorem.

(e)

To determine

The value of ( −12+32i)18 in standard form by using De Moivre’s Theorem.

(f)

To determine

The value of ( 2+2i)9 in standard form by using De Moivre’s Theorem.

(g)

To determine

The value of (1− 3 i)8 in standard form by using De Moivre’s Theorem.

(h)

To determine

The value of ( −12+32i)12 in standard form by using De Moivre’s Theorem.

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Let 2 A = 4 3 -4 0 1 (a) Show that v = eigenvalue. () is an eigenvector of A and find the corresponding (b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a) may be useful. (c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces. (d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.

(c) Let 6 0 0 A = -10 4 8 5 1 2 (i) Find the characteristic polynomial of A and factorise it. (ii) Determine all eigenvalues of A and find bases for the corresponding eigenspaces. (iii) Is A diagonalisable? Give reasons for your answer.

most 2, and let Let P2 denote the vector space of polynomials of degree at D: P2➡ P2 be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative p'(t) 2at+b, that is, D(p) = p'. (a) Prove that D is a linear transformation. (b) Find a basis for the kernel ker(D) of the linear transformation D and compute its nullity. (c) Find a basis for the image im(D) of the linear transformation D and compute its rank. (d) Verify that the Rank-Nullity Theorem holds for the linear transformation D. (e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.

Answer 2

Question

Chapter 7.3, Problem 3E

(a)

To determine

The value of (3+i)7 in standard form by using De Moivre’s Theorem.

(b)

To determine

The value of (32+12i)21 in standard form by using De Moivre’s Theorem.

(c)

To determine

The value of (− 3+i)10 in standard form by using De Moivre’s Theorem.

(d)

To determine

The value of ( 32−12i)18 in standard form by using De Moivre’s Theorem.

(e)

To determine

The value of ( −12+32i)18 in standard form by using De Moivre’s Theorem.

(f)

To determine

The value of ( 2+2i)9 in standard form by using De Moivre’s Theorem.

(g)

To determine

The value of (1− 3 i)8 in standard form by using De Moivre’s Theorem.

(h)

To determine

The value of ( −12+32i)12 in standard form by using De Moivre’s Theorem.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 7.3, Problem 3E

(a)

To determine

The value of (3+i)7 in standard form by using De Moivre’s Theorem.

(b)

To determine

The value of (32+12i)21 in standard form by using De Moivre’s Theorem.

(c)

To determine

The value of (− 3+i)10 in standard form by using De Moivre’s Theorem.

(d)

To determine

The value of ( 32−12i)18 in standard form by using De Moivre’s Theorem.

(e)

To determine

The value of ( −12+32i)18 in standard form by using De Moivre’s Theorem.

(f)

To determine

The value of ( 2+2i)9 in standard form by using De Moivre’s Theorem.

(g)

To determine

The value of (1− 3 i)8 in standard form by using De Moivre’s Theorem.

(h)

To determine

The value of ( −12+32i)12 in standard form by using De Moivre’s Theorem.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 7.3, Problem 3E

(a)

To determine

The value of (3+i)7 in standard form by using De Moivre’s Theorem.

(b)

To determine

The value of (32+12i)21 in standard form by using De Moivre’s Theorem.

(c)

To determine

The value of (− 3+i)10 in standard form by using De Moivre’s Theorem.

(d)

To determine

The value of ( 32−12i)18 in standard form by using De Moivre’s Theorem.

(e)

To determine

The value of ( −12+32i)18 in standard form by using De Moivre’s Theorem.

(f)

To determine

The value of ( 2+2i)9 in standard form by using De Moivre’s Theorem.

(g)

To determine

The value of (1− 3 i)8 in standard form by using De Moivre’s Theorem.

(h)

To determine

The value of ( −12+32i)12 in standard form by using De Moivre’s Theorem.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 5

Chapter 7.3, Problem 3E

(a)

To determine

The value of (3+i)7 in standard form by using De Moivre’s Theorem.

(b)

To determine

The value of (32+12i)21 in standard form by using De Moivre’s Theorem.

(c)

To determine

The value of (− 3+i)10 in standard form by using De Moivre’s Theorem.

(d)

To determine

The value of ( 32−12i)18 in standard form by using De Moivre’s Theorem.

(e)

To determine

The value of ( −12+32i)18 in standard form by using De Moivre’s Theorem.

(f)

To determine

The value of ( 2+2i)9 in standard form by using De Moivre’s Theorem.

(g)

To determine

The value of (1− 3 i)8 in standard form by using De Moivre’s Theorem.

(h)

To determine

The value of ( −12+32i)12 in standard form by using De Moivre’s Theorem.

Answer 6

Chapter 7.3, Problem 3E

(a)

To determine

The value of (3+i)7 in standard form by using De Moivre’s Theorem.

Answer 7

Chapter 7.3, Problem 3E

(a)

To determine

The value of (3+i)7 in standard form by using De Moivre’s Theorem.

Answer 8

(b)

To determine

The value of (32+12i)21 in standard form by using De Moivre’s Theorem.

Answer 9

(b)

To determine

The value of (32+12i)21 in standard form by using De Moivre’s Theorem.

Answer 10

(c)

To determine

The value of (− 3+i)10 in standard form by using De Moivre’s Theorem.

Answer 11

(c)

To determine

The value of (− 3+i)10 in standard form by using De Moivre’s Theorem.

Answer 12

(d)

To determine

The value of ( 32−12i)18 in standard form by using De Moivre’s Theorem.

Answer 13

(d)

To determine

The value of ( 32−12i)18 in standard form by using De Moivre’s Theorem.

Answer 14

(e)

To determine

The value of ( −12+32i)18 in standard form by using De Moivre’s Theorem.

Answer 15

(e)

To determine

The value of ( −12+32i)18 in standard form by using De Moivre’s Theorem.

Answer 16

(f)

To determine

The value of ( 2+2i)9 in standard form by using De Moivre’s Theorem.

Answer 17

(f)

To determine

The value of ( 2+2i)9 in standard form by using De Moivre’s Theorem.

Answer 18

(g)

To determine

The value of (1− 3 i)8 in standard form by using De Moivre’s Theorem.

Answer 19

(g)

To determine

The value of (1− 3 i)8 in standard form by using De Moivre’s Theorem.

Answer 20

(h)

To determine

The value of ( −12+32i)12 in standard form by using De Moivre’s Theorem.

Answer 21

(h)

To determine

The value of ( −12+32i)12 in standard form by using De Moivre’s Theorem.

Answer 22

Expert Solution & Answer

Answer 23

Expert Solution & Answer

Answer 24

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Answer 25

The value of ( 3 + i ) 7 in standard form by using De Moivre’s Theorem.

Solution Summary: The author explains De Moivre's Theorem to determine the value of (sqrt3+i)7 in standard form.

The value of (3+i)7 in standard form by using De Moivre’s Theorem.

The value of (32+12i)21 in standard form by using De Moivre’s Theorem.

The value of (− 3+i)10 in standard form by using De Moivre’s Theorem.

The value of ( 32−12i)18 in standard form by using De Moivre’s Theorem.

The value of ( −12+32i)18 in standard form by using De Moivre’s Theorem.

The value of ( 2+2i)9 in standard form by using De Moivre’s Theorem.

The value of (1− 3 i)8 in standard form by using De Moivre’s Theorem.

The value of ( −12+32i)12 in standard form by using De Moivre’s Theorem.

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Chapter 7 Solutions