Question

Write the quotient in standard form. (9-13i)/(1-2i)

Answer 1

To write (9-13i)/(1-2i) in standard form, multiply by the complex conjugate of the denominator (1+2i) and simplify the expression. The final quotient in standard form is -17/5 + i.

Step-by-step explanation:

To express the quotient (9-13i)/(1-2i) in standard form, we need to eliminate the complex number in the denominator. We can do this by multiplying both the numerator and denominator by the complex conjugate of the denominator, which in this case is (1+2i). This process, known as rationalizing the denominator, ensures that the denominator becomes a real number.

Here’s the step-by-step calculation:

1. Multiply the numerator and denominator by the complex conjugate of the denominator:
   (9-13i)(1+2i)/(1-2i)(1+2i)
2. Apply the distributive property in the numerator:
   (9*1 + 9*2i - 13i*1 - 13i*2i)/(1 - 4i + 4i - 4i^2)
3. Calculate the products and simplify:
   (9 + 18i - 13i - 26)/(1 + 4)
4. Combine like terms in the numerator and note that i^2 = -1:
   (-17 + 5i)/5
5. Separate the real and imaginary parts by dividing each term by the real number in the denominator:
   -17/5 + (5i/5)
6. Write the final answer in standard form:
   -17/5 + i

This standard form is a+bi where a is the real part and b is the coefficient of the imaginary part i.