Example 6

Transcript

Example 6 Express the following in the form a + ib (5 + √2 i )/(1 − √2 i) (5 + √2 𝑖 )/(1 − √2 𝑖) Rationalizing = (( 5 + √2 𝑖) )/((1 − √2 𝑖 ) ) × ((1 + √2 𝑖))/((1 + √2 𝑖) ) = (( 5 + √2 𝑖) (1 + √2 𝑖 ))/((1 − √2 𝑖 ) (1 + √2 𝑖) ) = ( ( 5 + √2 𝑖 ) ( 1 +√2 𝑖))/(( 1 )2 − (√2 𝑖 )2) = ( ( 5 + √2 𝑖 ) ( 1 +√2 𝑖) )/(1 − 2𝑖 2) = ( 5 × 1 + 5 × √2 𝑖 + √2 𝑖 × 1 + √2 𝑖 × √2 𝑖)/(1 −2𝑖2) = ( 5 + 5 √2 𝑖 + √2 𝑖 + 2𝑖2)/(1 −2𝑖2) Putting i 2 = −1 = ( 5 + 5√2 𝑖 + √2 𝑖 + 2 ( − 1 ))/(1 −2 ( −1 )) = ( 5 + 5 √2 𝑖 + √2 𝑖 − 2)/(1 + 2) = ( 5 −2 +5 √2 𝑖 +√2 𝑖)/3 = ( 3 + 6 √2 𝑖 )/3 = ( 3 ( 1+2 √2 𝑖 ))/3 = 1 + 2 √2 𝑖 Example 6 Express the following in the form a+ ib (ii) 𝑖^(−35) 𝑖^(−35) = 1/𝑖^35 = 1/(〖(𝑖〗^34) × 𝑖) = 1/((𝑖^2 )^17 × 𝑖 ) Putting i 2 = − 1 = 1/(( −1 ) 17 × 𝑖) = 1/("−" 1× 𝑖) = 1/(− 𝑖) Multiplying and divided by – 𝑖 = 1/(− 𝑖) × (− 𝑖)/(− 𝑖) = (− 𝑖 )/( 𝑖^2 ) Putting i 2 = − 1 = (− 𝑖 )/( 𝑖 ) = 𝑖 = 0 + 𝑖