How do i rationalize the expression (4 + 3i)/(2 - i)?

Question

please help

Answer 1

(4 + 3i) / (2 - i) =

(4 + 3i(2 + i) / (2 - i)(2 + i) =

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Multiply tne numerator

(4 + 3i)((2 + i) =

8 + 6i + 4i + 3i² =

8 + 10i + 3i² =

8 + 10i + 3(- 1) =

8 + 10i + (- 3) =

8 + 10i - 3

5 + 10i
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Multiply the denominator

(2 - i)(2 + i) =

4 - 2i + 2i - i² =

4 - i²

4 - (-1)

4 + 1

5

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Put the numerator and denominator together

5 + 10i / 5 =

1 + 2i / 1=

1 + 2i

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Answer 2

Multiply the numerator and denominator by the complex conjugate of the denominator. The denominator is 2 - i, and its complex conjugate is 2 + i. Multiply the fraction by (2 + i)/(2 + i), which is equal to 1 and therefore doesn't change the value.

(2 - i)(2 + i) = 4 - i^2 = 4 - (-1) = 4 + 1 = 5. So as you can see, the denominator will be 5, a real number. The new numerator, (4 + 3i)(2 + 1), a calculation I leave to you, will probably still be complex, but once you have the numerator in the form a + bi, you can just divide a and b by your new denominator of 5 so that the whole expression is equal to (a/5) + (b/5)i, much nicer than the original complex fraction.

Answer 3

to locate the price of ?(2i), i.e. sq. root of (2i): on the Argand Diagram, the point 2i is on the y-axis, so ? = ?/2 and the size is two. each and every authentic or complicated volume has n nth roots, so we think 2 sq. roots the following. instruct 2i in trig/polar style: 2(cos(?/2) + isin(?/2)). making use of De Moivre's Theorem, the roots are for this reason = ?2 (cos[(?/2 + 2?k)/2] + isin[(?/2 + 2?k)/2]), for ok=0, a million Roots are for this reason: ?2 (cos[?/4] + isin[?/4]) = a million+ i ?2 (cos[5?/4] + isin[5?/4]) = -a million-i you may do an similar to simplify a million-?(3i) back you receives 2 a threat complicated numbers.

Answer 4

hi,

4+3i / 2-i * 2+i/2+i

(4+3i)(2+i)/ (2+i)(2-i) = 8 +4i +6i -3 /4 +1

5 + 10i /5 = 1 +2i

Answer 5

(4 + 3ì)/(2 - ì) = (multiply top and bottom by it's conjugate).
(4 + 3ì)*(2 + ì)/(2 - ì)*(2 + ì) =
[8 + 4ì + 6ì + 3ì²] / [4 + 2ì - 2ì - ì²] =
[8 + 10ì + 3ì²] / [4 - ì²] =

But ì² = -1

[8 + 10ì + 3(-1)] / [4 - (-1)] =
5 + 10ì / 5 =
= 1 + 2ì

Answer 6

1+2i
Please give me best answer thanks!