Let z = 8 + 3i and w = 7 + 2i, find z/w and z.w

For the given complex number, z = 8 + 3i, and w = 7 + 2i, value for z/w is, 62/53 + 5/53i and the value of z.w is, 50 + 5i. The detailed solution for the same is given below.

For z = 8 + 3i and w = 7 + 2i, find z/w. Determine (8 + 3i) (7 + 2i) and simplify as much as possible, writing the result in the form a + bi, where a and b are real numbers. 

Solution:

Given:

• z = 8 + 3 i

w = 7 + 2i

To find z/w,

= (8 + 3i )/(7 + 2i)

Multiplying the numerator and denominator with the conjugate of denominators.

= {(8 + 3i) / (7 + 2i) } × {(7 – 2i) / (7 – 2i)}

= {(8 + 3i)(7 – 2i)} / {(7 + 2i)(7 – 2i)}

= {56 – 16i + 21i – 6i²} / {(7)² – (2i)²}

= {56 + 5i – 6(-1)} / {49 – 4(i² )}

= {56 + 5i +6} / {49 +4}

= (62 +5i) / 53

z/w = 62/53 + 5/53i

Now, to find z.w,

= (8 + 3i).(7 + 2i)

= {56 + 16i + 21i + 6i²}

= {56 + 37i + 6(-1)}

= {56 + 5i – 6}

= 50 + 5i

Question 1: Express (2 – i)/(1 + i) in standard form?

Solution:

Given:

• (2 – i)/(1 + i)

Multiplying the numerator and denominator with the conjugate of denominators,

= {(2 – i)/(1 + i) × (1 – i)/(1 – i)}

= {(2 – i)(1 – i)} / {(1)² – (i)²}

= {2 – 2i – i – i²} / (1-i²)

= {2 – 3i – (-1)} / (1+1)

= ( 3 – 3i) / 2

= 3/2 – 3/ 2 i

Question 2:  Simplify in form of a + ib, (-5i)(2/8i)

Solution:

Given:

• (-5i)(2/8i)

= (- 10/8 )i²

= (- 10/8 )(-1)

= 10/8 + 0i =  5/4 + 0i

Question 3:  For z = 3 + 3i and w = 5 + 2i, find z/w. That is, determine (3 + 3i) (5 + 2i) and simplify as much as possible, writing the result in the form a+bi, where a and b are real numbers.

Solution: 

Given:

• z = 3 + 3i
• w = 5 + 2i

To find z/w

= (3 + 3i )/ (5 + 2i)

Multiplying the numerator and denominator with the conjugate of denominators.

= {(3 + 3i) / (5 + 2i)} × {(5 – 2i) / (5 – 2i)}

= {(3 + 3i)(5 – 2i)} / {(5 + 2i)(5 – 2i)}

= {15 – 6i +15i – 6i²} / {(5)² – (2i)²}

= {15 + 9i – 6(-1)} / {49 – 4(i² )}

= {15 + 9i + 6} / {49 + 4}

= (21 + 9i) / 53

 z/w = 21/53 + 9/53i

Now (3 + 3i) (5 + 2i)

= {15 + 6i +15i + 6i²}

= {15 + 21i + 6(-1)}

= {15 + 21i – 6}

= 9 + 21i

Question 4: Perform the indicated operation and write the answer in standard form: (2 – 14i)(2 + 14i)

Solution: 

Given:

• (2 – 14i)(2 + 14i)

= {(4 + 28i – 28i -196i² )}

= ( 4 + 196)

= 200 + 0i

Question 5: Perform the indicated operation and write the answer in standard form: (7 + 2i) × (5 – 4i)

Solution: 

(7 + 2i) × (5 – 4i)

= (35 – 28i + 10i – 8i²)

= {35 – 18i – 8(-1)} 

= 35 – 18i + 8

= 43 – 18i

Question 6: Simplify -3 + 8i?

Solution: 

Multiplicative inverse of a complex number z is simply 1/z.

It is denoted as: 1 / z  or  z – 1 (Inverse of z)

Here z = -3 + 8i

Therefore z = 1/z

= 1 / (-3 + 8i)

Now rationalizing,

= 1/(-3 + 8i) × (-3 – 8i)/(-3 – 8i)

= (-3 – 8i ) / {(-3)² – 8²i²}

= (-3 – 8i) / {9 + 64}

= (-3 – 8i)/ (73)

= -3/73 – 8i/73

Question 7: What is the solution to the following problem, (-3i)(9i)(-1).

Solution: 

Given:

• (-3i)(9i)(-1)

= -3i × 9i × (-1)

= -27i² × -1 {i² = -1}

= -27 (-1) × -1

= 27 × -1

= -27 + 0i