What is the product of 36‾√cis (π8) and 25‾√cis (7π6) ?

The product of

36√cis (π/8) and 25√cis (7π/6)

is

(225√2) √[√ (2 + √2) + i√ (2 - √2) ][√ (3 (-1 + i)) ]

Step-by-step explanation:

First note that

cis (π/8) = cos (π/8) + isin (π/8)

cis (7π/6) = cos (7π/6) + isin (7π/6)

cos (π/8) = (1/2) √ (2 + √2)

sin (π/8) = (1/2) √ (2 - √2)

36√cis (π/8) = (36/√2) √[√ (2 + √2) + i√ (2 - √2) ]

cos (7π/6) = - (1/2) √3

sin (7π/6) = (1/2) √3

25√cis (7π/6) = (25/2) √3 (-1 + i)

The product,

36√cis (π/8) * 25√cis (7π/6)

= (36/√2) √[√ (2 + √2) + i√ (2 - √2) ] * (25/2) √3 (-1 + i)

= (225√2) √[√ (2 + √2) + i√ (2 - √2) ][√ (3 (-1 + i)) ]