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3 √
√
√
3. Let z = − 3 2 3 + 3 i and w = 3 2 − 3i 2. Compute the following. Express your answers in
2
polar form using the principal argument. (a) zw
(b) (c) z
w (d) w
z
z4 (e) w3
(f) z 5 w2 4. Find the following complex roots. Express your answers in polar using the principal argument
and then convert them into rectangular form.
(a) the three cube roots of z = i
(b) the six sixth roots of z = 64
(c) the two square roots of 5
2 − √
53
2i 5. Use the Sum and Diﬀerence Identities in Theorem 10.16 or √ Half Angle Identities in
the
√
Theorem 10.19 to express the three cube roots of z = 2 + i 2 in rectangular form. (See
Example 11.7.4, number 3.)
6. Use a calculator to approximate the ﬁve ﬁfth roots of 1. (See Example 11.7.4, number 4.)
7. According to Theorem 3.16 in Section 3.4, the polynomial p(x) = x4 + 4 can be factored
into the product linear and irreducible quadratic factors. In Exercise 13 in Section 8.7, we
showed you how to factor this polynomial into the product of two irreducible quadratic factors
using a system of non-linear equations. Now that we can compute the complex fourth roots
of −4 directly, we can simply apply the Complex Factorization Theorem, Theorem 3.14, to
obtain the linear factorization p(x) = (x − (1 + i))(x − (1 − i))(x − (−1 + i))(x − (−1 − i)).
By multiplying the...

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