Question

evaluate the radical expression and express the result in a+bi form

Answer 1

\[(3-\sqrt{-5}) (1+\sqrt{-1})\]

Answer 2

well you need to distribute so it is

\[3(1 + \sqrt{-1})-\sqrt{-5}(1 + \sqrt{-1})\]

what do you think the next line of working is..?

Answer 3

\[3(1+i)-i \sqrt{5}(1+i)\]

Answer 4

?

Answer 5

good,
now distribute the 3, and the -i√5

Answer 6

\[(3+3i)(-i \sqrt{5}+1\sqrt{5}) ?\]

Answer 7

\[(3-\sqrt{-5}) (1+\sqrt{-1}) \]
\[(3-\sqrt{5}i) (1+i)\]
now expand.

Answer 8

all negatives in the radical should be pulled out first.. so that square root of -5 should be square root of 5 i

Answer 9

\[3(1+i)-i \sqrt{5}(1+i)\\=(3+3i)+(-i \sqrt{5}-\sqrt{5}i\times i) \]

Answer 10

\[(3-\sqrt{5}i) (1+i) \]
\[3+3i-\sqrt{5}i-\sqrt{5}(i)(i)\]
note \[i^2 = -1 \]

Answer 11

typed too fast... -1 inside the square root is just an i

Answer 12

\[(3+\sqrt{5})+(3-\sqrt{5})i\]

Answer 13

is that right?

Answer 14

hold on.

Answer 15

\[3+3i-\sqrt{5}i-\sqrt{5}(-1)\]
\[3+3i-\sqrt{5}i+\sqrt{5}\]
\[3+\sqrt{5}+3i-\sqrt{5}i\]

\[\[3+\sqrt{5}+i(3-\sqrt{5})\]\] yeah it's correct

Answer 16

my i is placed differently, but it shouldn't matter because we still have a+bi only our a =\[3+\sqrt{5}\] and b = \[3-\sqrt{5}\]

Answer 17

thank you so much!

Answer 18

it's best to convert all negatives in the square root to i's first and if it's a perfect square like \[\sqrt{-1} \] just take the square root and add an i
\[\sqrt{-1} \rightarrow i \]
similarly for \[\sqrt{-5} \rightarrow \sqrt{5}i\]

Answer 19

but 5 isn't a perfect square so leave it in the radical and only the negative pops out of the radical and becomes i

Answer 20

negatives inside the radical produce imaginary results.

Answer 21

then use foil and i^2 = -1 ... simplify until a+bi or ai+b form is achieved.