Question

Find the modules of(1-3i)¹⁰

Answer 1

(1-3i)¹⁰  = [(1-3i)²]⁵ = [ 1 - 9 -6i]⁵ = [ -8 -6i]⁵ = (-2)⁵ [ 4 +3i]⁵

= (-2)⁵ [ 4 +3i]²[ 4 +3i]²[ 4 +3i] =  (-2)⁵ [ 16-9 + 24i ][ 16-9 + 24i ][ 4 +3i] 

=  -32[ 7 + 24i ][ 7 + 24i ][ 4 +3i] =  -32[ 49 - 576 + 336i ][ 4 +3i]

= -32[ - 527 + 336i ][ 4 +3i] = -32[ - 2108 - 1008 - 1581i + 1344i  ] = 32 [ 3116 + 237i ]

Modulus of (1-3i)¹⁰ = 32 { sqrt (  3116* 3116 + 237*237 ) } = 32 { 3125 } = 100000