Question

Write the quotient in (3+i)/(3-i) standard form.

Answer 1

The quotient of (3+i)/(3-i) in standard form is 0.8 + 0.6i. This result is obtained by multiplying the numerator and denominator by the conjugate of the denominator, simplifying, and then dividing each term to get the standard form.

Step-by-step explanation:

To write the quotient in standard form for the complex number (3+i)/(3-i), we need to rationalize the denominator. We do this by multiplying the top and bottom of the fraction by the conjugate of the denominator. In this case, the conjugate of (3-i) is (3+i).

Here are the steps:

1. Multiply both the numerator and denominator by the conjugate of the denominator: (3+i) * (3+i) / (3-i) * (3+i).
2. Simplify the expression by using the distributive property (FOIL method) in the numerator and denominator to get: (9 + 3i + 3i + i^2) / (9 - i^2).
3. Knowing that i^2 = -1, simplify the equation further to: (9 + 6i - 1) / (9 + 1) which equals 8 + 6i / 10.
4. Divide each term in the numerator by the denominator to get 0.8 + 0.6i, which is the quotient in standard form.

The standard form of the quotient (3+i)/(3-i) is 0.8 + 0.6i.