Squaring / Square Roots / Radicals - ACT Math

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

In equation , is the real component and is the imaginary component (designated by ).

In equation , is the real component and is the imaginary component.

When added,

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

First, distribute:

Then, group the real and imaginary components:

Solve to get:

When adding or subtracting complex numbers, the real terms are additive/subtractive, and so are the nonreal terms.

Complex numbers take the form a + bi, where a is the real term in the complex number and bi is the nonreal (imaginary) term in the complex number. Taking this, we can see that for the real number 8, we can rewrite the number as , where represents the (zero-sum) non-real portion of the complex number.

Thus, any real number can be added to any complex number simply by considering the nonreal portion of the number to be .

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Thus, to balance the equation, add like terms on the left side.

This problem can be solved in a way similar to other kinds of division problems (with binomials, for example). We need to get the imaginary number out of the denominator, so we will multiply the denominator by its conjugate and multiply the top by it as well to preserve the number's value.

Then, recall by definition, so we can simplify this further:

This is as far as we can simplify, so it is our final answer.

Multiply both numberator and denominator by :

First, divide 100 by as follows:

Now dvide this result by :

First, divide 100 by as follows:

Now, divide this by :

First, evaluate :

Now divide this into :

First, evaluate using the square pattern:

Divide this into :

This problem can be solved very similarly to a binomial such as . In this case, both the real and nonreal terms in the complex number are eligible to be divided by the real divisor.

, so

Solving this problem using a conjugate is just like conjugating a binomial to simplify a denominator.

Multiply both terms by the denominator's conjugate.

Simplify. Note .

Combine and simplify.

Simplify the numerator.

The prime denominator prevents further simplifying.

Thus, .

Simplify the base of 9 and 27 in order to have a common base.

(3x)(9)=272

= (3x)(32)=(33)2

=(3x+2)=36

Therefore:

x+2=6

x=4

The terms of have as their greatest common factor, so

is a prime polynomial.

Of the five choices, only is a factor.

First, break down the component parts of the square root:

Combine like terms in a way that will let you pull some of them out from underneath the square root symbol:

Pull out the terms with even exponents and simplify:

First, break down the components of the square root:

Combine like terms. Remember, when multiplying exponents, add them together:

Factor out the common factor of :

Factor the :

Combine the factored with the :

Now, you can pull out from underneath the square root sign as :

When simplifying a square root, consider the factors of each of its component parts:

Combine like terms:

Remove the common factor, :

Pull the outside of the equation as :

The easiest way to approach this problem is to break everything into exponents. is equal to and 27 is equal to . Therefore, the expression can be broken down into . When you cancel out all the terms, you get , which equals .