# Operations on Imaginary Numbers

PLEASE NOTE THAT YOU CANNOT USE A CALCULATOR ON THE ACCUPLACER -

ELEMENTARY ALGEBRA TEST! YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS

WITHOUT A CALCULATOR!

Simplifying Square- Root Radicals Containing Perfect Square Factors

Some radicands are perfect squares , whereas others are not. In the later case it
is often

useful to "simplify" the radicand IF it contains a perfect-square factor!

•If the radicand is a perfect square, find its square root.

•If the radicand is not a perfect square, try to factor the radicand so that one
factor is

the largest possible perfect square factor. Place the square root of this factor

outside the radical and keep the other factors inside (or under) the radical.
There is

an assumed multiplication sign between the number outside the radical and

and the radical expression.

•If the radicand is not a perfect square and it does not have a perfect square
factor,

then it is simplified.

**Problem 1:**

Simplify .

The word "simplify" takes on many meanings in mathematics. Often you must

figure out its meaning from the mathematical expression you are asked to

"simplify." Here we are asked to "simplify" instead of finding the square root.

Since the radicand is a perfect square, we can say
. This is NOT an irrational

number.

**Problem 2:
**

Simplify .

The word "simplify" takes on many meanings in mathematics. Often you must

figure out its meaning from the mathematical expression you are asked to

"simplify." Here we are asked to "simplify" instead of factoring out perfect-square

factors.

Here the radicand is not a perfect square. However, we
know that we can write the

radicand as a product as follows :

Since we know that 16 is a perfect square and it is also the largest perfect
square factor

of the radicand, we can now simplify the square root.

That is, in simplified form.

Please note that means 4 times the
.

**Problem 3:
**

Simplify .

Here the radicand is not a perfect square. However, we know that we can write the

radicand as a product as follows:

Since we know that 9 is a perfect square and it is also the largest perfect square factor of

the radicand, we can now simplify the square root.

That is, in simplified form.

**Problem 4:**

Simplify.

Here the radicand is not a perfect square. However, we know that we can write the

radicand as a product as follows:

Since we know that 4 is a perfect square and it is also
the largest perfect square factor of

the radicand, we can now simplify the square root.

That is, in simplified form.

**Problem 5:**

Simplify .

Here the radicand is not a perfect square. However, we know that we can write
the

radicand as a product as follows:

Since we know that 4 is a perfect square and it is also the largest perfect
square factor of

the radicand, we can now simplify the square root.

That is, in simplified form.

**Problem 6:**

Simplify .

Here the radicand is not a perfect square. However, we know that we can write
the

radicand as a product as follows:

Since we know that 9 is a perfect square and it is also the largest perfect
square factor of

the radicand, we can now simplify the square root.

That is, in simplified form.

** Adding and Subtracting Square -Root Radicals**

Add or subtract the coefficients of like radicals .

Use the common radical as a factor in the solution.

**Problem 7:**

Add and simplify, if possible.

The word "simplify" takes on many meanings in mathematics. Often you must

figure out its meaning from the mathematical expression you are asked to

"simplify." Here we are asked to "simplify" instead of factoring perfect-square

factors out of the sum.

Since both terms have the same radical we can add their coefficient to get

**Problem 8:**

Add and simplify, if possible.

Please note that there is an implied coefficient of 1 in front of
. It is NOT customary to

write it. However, it must be used in addition and subtraction!

Since both terms have the same radical we can add their coefficient to get

**Problem 9:**

Combine the "like" radicals in and simplify,
if

possible.

The word "simplify" takes on many meanings in mathematics. Often you must

figure out its meaning from the mathematical expression you are asked to

"simplify." Here we are asked to "simplify" instead of factoring perfect-square

factors out of the result

Let's rearrange the expression as follows and then combine the terms with like
radicals.

That is,

Pleasenote that we cannot add the coefficients of the final difference! There
are no like

radicals and the radicands are prime numbers.

**Problem 10:**

Subtract and simplify, if possible.

The word "simplify" takes on many meanings in mathematics. Often you must

figure out its meaning from the mathematical expression you are asked to

"simplify." Here we are asked to "simplify" instead of factoring perfect-square

factors out of the difference.

No simplification can take place. The only thing that could be done is to change
the

radical to a rounded decimal number and then subtract.

**
Problem 11:**

Subtract and simplify, if possible.

No simplification can take place. The only thing that could be done is to change the

radicals to rounded decimal numbers and then subtract.

**Problem 12:**

Add and simplify, if possible.

There are no like radicals. However, 20 is NOT a prime number, therefore, we can

simplify the second radical to see if we end up with a sum of like radicals .

and we find that we DO have like radicals
when simplifying!

Now we can write

**Problem 13:**

Subtract and simplify, if possible.

There are no like radicals. However, 32 is NOT a prime number, therefore, we can

simplify the second radical to see if we end up with a difference of like
radicals.

and we find that we do have like radicals
when simplifying!

Now we can write

Please note that there is an implied coefficient of 1 in front of
. It is NOT customary to

write it. However, it must be used in addition and subtraction!

**Problem 14:**

Subtract and simplify, if possible.

There are no like radicals. However, 18 is NOT a prime number, therefore, we can

simplify the second radical to see if we end up with a sum of like radicals.

and we find that we still DO NOT have like
radicals when

simplifying!

However, we might want to write

Please note that we cannot add the coefficients of the
final difference! There are no like

radicals and the radicands are now prime numbers.

**Multiplying Square-Root Radicals**

•Multiply the coefficients.

•Multiply the radicands.

•Simplify, if possible.

Symbolically , this can be stated as .

**
Problem 15:**

Multiply and simplify, if possible.

The word "simplify" takes on many meanings in mathematics. Often you must

figure out its meaning from the mathematical expression you are asked to

"simplify." Here we are asked to "simplify" instead of factoring perfect-square

factors out of the product.

The coefficients are both 1. But we can multiply the
radicands as follows:

. The radical cannot be simplified because 91
is a prime number.

**Problem 16:**

Multiply and simplify, if possible.

The coefficients are both 1. But we can multiply the radicands as follows:

. The product can be reduced!

That is, which cannot be simplified.

**Problem 17:**

Multiply and simplify, if possible.

The coefficients are 3 and 2. Therefore, they need to be multiplied as well as
the

radicands.

That is . The radical
cannot be simplified because 35

is a prime number.

**Problem 18:**

Multiply and simplify, if possible.

The coefficients are 4 and 2. Therefore, they need to be multiplied as well as
the

radicands.

That is . The radical is NOT a prime number
and

we may be able to simplify it.

That is,

**Problem 19:**

Multiply and simplify, if possible.

and

NOTE: Whenever you multiply a radical by itself, the
solution is ALWAYS the

radicand!

**Problem 20:**

Multiply and simplify, if possible.

Here we have to use the** FOIL** process. Remember that FOIL stands for First
Product,

Outer Product, Inner Product, and Last Product.

That is,

Please note that the two middle products have like radicands that is why we
added their

coefficients!

**Problem 21:**

Multiply and simplify, if possible.

Here we have to use the **FOIL** process again.

Please note the following:

Since the two middle products have the same value, they cancel each other out
when one

is subtracted from the other!

Therefore, we get

** Dividing Square -Root Radicals**

Divide the coefficients.

Divide the radicands.

Simplify, if possible.

Symbolically, this can be stated as .

**
Problem 22:**

Divide and simplify, if possible.

The word "simplify" takes on many meanings in mathematics. Often you must

figure out its meaning from the mathematical expression you are asked to

"simplify." Here we are asked to "simplify" instead of factoring perfect-square

factors out of the quotient.

The radical cannot be simplified because 6 does not contain a perfectsquare

factor.

**Problem 23:**

Divide and simplify, if possible.

The coefficient 3 in the numerator and the denominator 6 can be divided (or
reduced)

because they are both outside the radical. NOTE: You cannot divide the 6 under
the

radical by the 6 in the denominator !

We get . The coefficient 1 in the numerator
is implied! The radical cannot be

simplified because 6 does not contain a perfect-square factor.

**Problem 24:**

Divide and simplify, if possible.

We get . The radical cannot be simplified
because 2 is a prime number.

** Rationalizing a Denominator containing an Irrational Square-Root Radical**

•Multiply the denominator by itself.

•To preserve the value of the fraction, multiply the numerator by the same
number.

•Simplify all radicals and reduce the resulting fraction, if possible.

NOTE: In Steps 1 and 2 above , we have actually multiplied the fraction by an

equivalent of the number 1!

**
Problem 25:**

Rationalize the denominator in and simplify, if possible.

The radical cannot be simplified because 11 is a prime number.

Remember that and
. Also .

NOTE: You cannot divide the 77 in the denominator by the 11 in the radical!

**Problem 26:**

Rationalize the denominator in and simplify,
if possible.

. The radical cannot be simplified because 6
does not contain a

perfect-square factor.

NOTE: You cannot divide the 6 under the radical by the 6 in the denominator !

However, the coefficient 4 in the numerator and the denominator 6 can be reduced

because they are both outside the radical.

That is, can be written as
. You cannot divide the 6 under the radical
by

the 3 in the denominator !

**Problem 27:**

Rationalize the denominator in and simplify,
if possible.

NOTE: You cannot divide the x under the radical by the x in the denominator !

**Problem 28:**

Rationalize the denominator in and simplify,
if possible.

The radical cannot be simplified because 39 does not contain a perfect-square
factor.

NOTE: You cannot divide the 39 under the radical by the 3 in the denominator !

**Problem 29:**

Rationalize the denominator in and simplify,
if possible.

The radical cannot be simplified because 2 is a prime number.

NOTE: You cannot divide the 2 under the radical by the 2 in the denominator !

**Problem 30:**

Rationalize the denominator in and simplify,
if possible.

First we need to be aware of the following:

Now we can rationalize the denominator as usual.

The radical cannot be simplified because 35 does not
contain a perfect-square factor.

NOTE: You cannot divide the 35 under the radical by the 5 in the denominator !

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