Questions: Simplify (5-3i)/(-5-3i) (-8+15i)/17 (-23+36i)/25 (-21+33i)/34 (-10+6)/17

Simplify (5-3i)/(-5-3i) (-8+15i)/17 (-23+36i)/25 (-21+33i)/34 (-10+6)/17
Transcript text: \[ \frac{5-3 i}{-5-3 i} \] Simplify $(-8+15 i) / 17$ $(-23+36 i) / 25$ $(-21+33 i) / 34$ $(-10+6) / 17$
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Solution

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Solution Steps

To simplify a complex fraction, multiply the numerator and the denominator by the conjugate of the denominator. This will eliminate the imaginary part in the denominator, allowing you to simplify the expression to a standard form \(a + bi\).

Step 1: Simplifying the First Expression

To simplify the expression \(\frac{5 - 3i}{-5 - 3i}\), we multiply the numerator and the denominator by the conjugate of the denominator, which is \(-5 + 3i\). This results in: \[ \frac{(5 - 3i)(-5 + 3i)}{(-5 - 3i)(-5 + 3i)} = \frac{-5 + 3i}{34} \] Thus, the simplified form is: \[ \frac{-8}{17} + \frac{15i}{17} \]

Step 2: Simplifying the Second Expression

The second expression is \(\frac{-8 + 15i}{17}\). This expression is already in its simplest form: \[ \frac{-8}{17} + \frac{15i}{17} \]

Step 3: Simplifying the Third Expression

For the third expression \(\frac{-23 + 36i}{25}\), it is also already in its simplest form: \[ \frac{-23}{25} + \frac{36i}{25} \]

Final Answer

The simplified forms of the expressions are:

  1. \(\frac{-8}{17} + \frac{15i}{17}\)
  2. \(\frac{-8}{17} + \frac{15i}{17}\)
  3. \(\frac{-23}{25} + \frac{36i}{25}\)

Thus, the final answers are: \[ \boxed{\frac{-8}{17} + \frac{15i}{17}}, \quad \boxed{\frac{-8}{17} + \frac{15i}{17}}, \quad \boxed{\frac{-23}{25} + \frac{36i}{25}} \]

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