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Give all answers in $a+b i$ form. $$\frac{3+i}{4-i \sqrt{2}}$$

   Give all answers in $a+b i$ form.
$$\frac{3+i}{4-i \sqrt{2}}$$

College Algebra

College Algebra

R. David Gustafson,… 11th Edition

Chapter 1, Problem 56

Instant Answer

Solved by Expert Amy Jiang

07/07/2020

Step 1

The conjugate of $4-i\sqrt{2}$ is $4+i\sqrt{2}$. So, we have: $$\frac{3+i}{4-i\sqrt{2}} \times \frac{4+i\sqrt{2}}{4+i\sqrt{2}}$$ Show more…

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Give all answers in $a+b i$ form. $$\frac{3+i}{4-i \sqrt{2}}$$

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Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any imaginary components from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by the complex conjugate of the denominator, thus producing a denominator that is a real number.

Complex Conjugate

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. For a complex number a + bi, the conjugate is a - bi. Multiplying a complex number by its conjugate yields a real number, a property that is particularly useful in the process of rationalizing denominators.

Complex Numbers

Complex numbers are numbers of the form a + bi, where a is the real part and b is the imaginary part, with i representing the square root of -1. They are fundamental in many areas of mathematics and engineering, providing a system where equations that do not have solutions in the real numbers can be solved.

Standard Form (a + bi) Representation

Expressing complex numbers in standard form (a + bi) clearly distinguishes the real part and the imaginary part, facilitating arithmetic operations like addition, subtraction, multiplication, and division, and making it easier to compare complex numbers.

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