Write in Standard Form 7/(4-14i)

Problem

7/(4−14*i)

Solution

1. Identify the goal to write the complex fraction in standard form a+b*i by eliminating the imaginary unit from the denominator.

2. Multiply the numerator and the denominator by the conjugate of the denominator, which is $4 + 14i$.

7/(4−14*i)⋅(4+14*i)/(4+14*i)

3. Distribute the numerator.

7*(4+14*i)=28+98*i

4. Expand the denominator using the difference of squares property (a−b*i)*(a+b*i)=a^2+b^2

(4−14*i)*(4+14*i)=4^2+14^2

16+196=212

5. Combine the results into a single fraction.

(28+98*i)/212

6. Simplify the fraction by dividing each term by the greatest common divisor, which is 2

(14+49*i)/106

7. Separate into real and imaginary parts to reach standard form.

14/106+49/106*i

8. Reduce the real part further by dividing by 2

7/53+49/106*i

Final Answer

7/(4−14*i)=7/53+49/106*i

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