Imaginary Numbers - Math

Begin by multiplying the numerator and denominator by the complement of the denominator:

Combine like terms:

Simplify. Remember that is equivalent to

The absolute value is a measure of the distance of a point from the origin. Using the pythagorean distance formula to calculate this distance.

Combine like terms. Treat as if it were any other variable.

Substitute to eliminate .

Simplify.

First, factor the term in the radical.

Now, we can simplify.

FOIL:

Since and are conmplex conjugates, they can be multiplied according to the following pattern:

Since and are conmplex conjugates, they can be multiplied according to the following pattern:

can be evaluated by dividing by 4 and noting the remainder. Since - that is, since dividing 45 by 4 yields remainder 1:

Recall the basic property of imaginary numbers, .

Keeping this in mind, .

Recall that .

Then, we have that .

Note that we used the power rule of exponents and the order of operations to simplify the exponent before multiplying by the coefficient.

Multiplying top and bottom by the complex conjugate eliminates i from the denominator

Use FOIL to multiply. We can treat as a variable for now, just as if it were an .

Combine like terms.

Now we can substitute for .

Simplify.

We will need to simplify the rational expression by removing imaginary terms from the denominator. Often in such problems, we want to multiply the numerator and denominator by the conjugate of the denominator, which will usually eliminate the imaginary term from the denominator.

In this problem, the denominator is . Remember that, in general, the conjugate of the complex number is equal to , where a and b are both nonzero constants. Thus, the conjugate of is equal to .

We need to multiply both the numerator and denominator of the fraction by .

Next, we use the FOIL method to simplify both the numerator and denominator. The FOIL method of binomial multiplication requires us to mutiply together the first, outside, inner, and last terms of each binomial and then sum them.

Now, we can start simplifying.

Use the fact that .

The answer is .

Begin by factoring the radical expression using imaginary numbers:

Now, multiply the factors:

Simplify. Remember that is equivalent to

Begin by completing the square:

Now, multiply the factors. Remember that is equivalent to

Solve the equation using complex numbers:

Begin by factoring the radical expression using imaginary numbers:

Now, multiply the factors:

Simplify. Remember that is equivalent to

Begin by simplifying the radicals using complex numbers:

Multiply the factors:

Simplify. Remember that is equivalent to .

Begin by completing the square:

Multiply the factors:

Simplify. Remember that is equivalent to .