How can we simplify (2+3i)(1-i)/(3-3i)?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to simplify the expression [(2 + 3i)(1 - i)]/(3 - 3i)

Multiply the numerator and denominator by the complex conjugate of (3 - 3i) which is (3 + 3i)

[(2 + 3i)(1 - i)]/(3 - 3i)

=> [(2 + 3i)(1 - i)(3 + 3i)]/(3 - 3i)(3 + 3i)

=> [(2 + 3i)(1 - i)(3 + 3i)]/(9 + 9)

Open the brackets of the numerator and multiply the terms

=> (2 + 3i - 2i + 3)(3 + 3i) / 18

=> (5 + i)(3 + 3i) / 18

=> (15 + 3i + 15i - 3) / 18

=> (12 + 18i) / 18

=> (2 + 3i) / 3

=> 2/3 + i

The required simplified form is 2/3 + i

hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

Given the complex number:

(2+3i)(1-i)/(3-3i)

First we will simplify the numerator by opening the brackets.

==> (2+3i)(1-i) = 2 - 2i + 3i -3i^2

But i^2 = -1

==> (2+3i)(1-i) = 2 +i +3 = 5+i

==> (5+i)/(3-3i)

Now we will multiply numerator and denominator by (3+3i)

==> (5+i)(3+3i)/(3-3i)(3+3i) = (15+15i+3i+3i^2)/(9+9)

                                             = (12+18i) / 18

                                              = 12/18 + i

                                               = 2/3 + i

==> Then the value of (2+3i)(1-i)/(3-3i) can be simplifies into the form (2/3) + i

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