**1: Express the given complex number in the form a + ib: (5i)(- ^{3}/_{5} i)**

Solution:

(5i)(-^{3}/_{5} i) = -5 x ^{3}/_{5} x i x i

= -3i^{2}

= -3(-1) = 3

**2: Express the given complex number in the form a + ib: i ^{9} + i ^{19}**

Solution:

i^{9} + i ^{19} = i^{4×2+1} + i^{4×4+3}

= (i^{4})^{2}.i + (i^{4})^{4}.i^{3}

[i^{4} = 1 and i^{3} = -1]

= 1 x i+1 x (-i)

= i + (-i)

= 0

**Express the given complex number on the form a + ib: i**^{-39}

Solution:

i^{-39} = i^{-4×9-3}

=(i^{4})^{-9} x i^{-3}

= (1)^{-9} x i^{-3}

= ^{1}/_{i}^{3}

= ^{1}/_{-I} x ^{i}/_{i} = ^{-i}/_{i}^{2} x ^{i}/_{i} = ^{-i}/_{i}^{2} = ^{-i}/_{-i} = i

**Express the given complex number in the form a + ib: 3(7 + 7i)+i(7+7i)**

Solution:

3(7 + 7i)+i(7+7i) = 21 + 21i + 7i +7i^{2}

= 21 + 28i + 7 x (-1)

= 14 + 28i

- E
**xpress the given comples number in the form a + ib: (1-i)-(-1+6i)**

Solution:

(1 – i )-(-1 + 6i) = 1 – i + 1 – 6i

= 2 – 7i

**Express the given complex number in the form a + ib: (**^{1}/_{5}+ i^{2}/_{5})-(4 + i^{5}/_{2})

Solution:

(^{1}/_{5} + i ^{2}/_{5})-(4 + I ^{5}/_{2}) = ^{1}/_{5} + ^{2}/_{5}i – 4 – i ^{5}/_{2}

=(^{1}/_{5} – 4) + i(^{2}/_{5} – ^{5}/_{2})

=-^{19}/_{5} + i(^{-21}/_{10})

= –^{19}/_{5} – ^{21}/_{10} i

**Express the given complex number in the form a + ib:**

**[( ^{1}/_{3} + i^{ 7}/_{3}) + (4 + i^{ 1}/_{3})] – (-^{4}/_{3} + i)**

Solution:

[(^{1}/_{3} + i^{ 7}/_{3}) + (4 + i^{ 1}/_{3})] – (-^{4}/_{3} + i) = ^{1}/_{3} + ^{ 7}/_{3 }i + 4 + ^{ 1}/_{3}i + ^{4}/_{3} – i

= ( ^{1}/_{3} + 4 + ^{4}/_{3}) + i(^{7}/_{3} + ^{1}/_{3} -1)

= ^{17}/_{3} + i ^{5}/_{3}

**Express the given complex number in the form a + ib: (1 – i)**^{4}

Solution:

(1 – i)^{4} = [(1 – i)^{2}]^{2}

= [1^{2} + i^{2} – 2i] = [1 – 1 – 2i]^{2} = (-2i)^{2} = (-2i) x (-2i) = 4i^{2} = -4

**Express the given complex number in the form a + ib: (**^{1}/_{3}+ 3i)^{2}

Solution:

(^{1}/_{3} + 3i)^{2} = (^{1}/_{3})^{3} + (3i)^{3} + 3(^{1}/_{3})(3i)(^{1}/_{3} + 3i)

= (^{1}/_{27}) + 27i^{3} + 3(^{1}/_{3})(3i)(^{1}/_{3} + 3i)

= ^{1}/_{27} + 27(-i) + i + 9i^{2}

= ^{1}/_{27} – 27i + i – 9

= (^{1}/_{27} – 9) + i(-27 + 1)

= ‑^{242}/_{27} – 26i

**express he given complex number in the form a + ib: (-2 –**^{1}/_{3}i)^{3}

Solution:

(-2 – ^{1}/_{3}i)^{3} = (-1)^{3} (2 + ^{1}/_{3} i)^{3}

= – [ 2^{3} + (^{1}/_{3})^{3} + 3(2)(^{i}/_{3})(2 + ^{i}/_{3})]

= – [ 8 + (i)^{3}/_{27} + 2i(2 + ^{i}/_{3})]

= – [ 8 – ^{i}/_{27} + 4i + 2i^{2}/_{3}]

= – [8 – ^{i}/_{27} + 4i – ^{2}/_{3}]

= – [^{22}/_{3} + ^{107}/_{27} i ]

= – ^{22}/_{3} – ^{107}/_{27} i

**Find the multiplicative inverse of the complex number 4 – 3i**

Solution:

Let z = 4 – 3i

Then ź = 4 + 3i and |z|^{2} = 4^{2} + (-3)^{2} = 16 + 9 = 25

Therefore the multiplicative inverse of 4 – 3i is given by

z^{-1} = ^{ź}/_{|z|}^{2} = ^{4 + 3i}/_{25 } = ^{4}/_{25} + ^{3}/_{25} i

**Find the multiplicative inverse of the complex number √5 + 3i**

Solution:

Let z = √5 + 3i

Then ź = √5 – 3i and |z|^{2} = (√5)^{2} + (3i)^{2} = 5 + 9 = 14

Therefore the multiplicative inverse of √5 + 3i

z^{-1} = ^{ź}/_{|z|}^{2} = ^{√5 + 3i}/_{14} = ^{√5}/_{14} + ^{3}/_{14} i

**Find the multiplicative inverse of the complex number –i**

**Solution:**

Let z = -i

Then ź = I and |z|^{2} = (i)^{2} = 1

Therefore the multiplicative inverse of –I is given by

z^{-1} = ^{ź}/_{|z|}^{2} = ^{i}/_{I} = i

**Express the following expression in the form of a + ib.**

^{(3 + i√5)(3 – i√5)}/_{(√3 + √2i) – (√3 – √2i) }

Solution:

^{(3 + i√5)(3 – i√5)}/_{(√3 + √2i) – (√3 – √2i) } = [3^{2 }– (i√5)^{2}] /_{(√3 + √2i – √3 + √2i )}

= (9 – 5i^{2})/2√2i

= ^{9+5}/_{2√2i} x ^{i}/_{i}

= ^{14i}/_{2√2i}^{2}

= ^{14i}/_{2√2i}^{2}

= ^{14i}/_{2√2(-1)}

= ^{-7i}/_{√2} x ^{√2}/_{√2} = ^{-7√2i}/_{2}