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Project 4: Simplifying Expressions with Radicals
Defintion
A radical, which we symbolize with the sign √ , is just the opposite of an exponent. In this class, the only radical we
will focus on is square roots, which are the opposite of squaring, or raising something to the second power. The square
root is just defined as the number (or expression) that would have to be squared in order to result in the number (or
expression) under the radical sign. In math, this same idea can be expressed this way:
↔ √
This means, for example, that the following must be true:
Remember that and here are standing in for ANY number or expression. So, for example:
√
1 1
Now you try! Simplify the following expressions:
√
5
2
2 3
What is “legal” when simplifying radicals?
Because radicals are just a different way of writing exponents, the same kinds of rules apply about what we can and
can’t do when simplifying radicals. Let’s see how each rule for exponent expressions applies to radicals (below we
assume that 0): Exponent expression Radical expression
This works because ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅ .
√ √ √
This works because ⋅
⋅
√
√
! This is true because
.
√ √ √ ! Radicals with different bases CANNOT be added, and a radical CANNOT be distributed over ADDITION!
This is true because we are simply combining like terms. For example, 2 3 5
√ √ √ Note: When these are added, the radical part stays the same, because this is always what happens with like
terms—we are simply counting up how many of the √ ’s there are.
These terms cannot be combined because they are NOT like terms—the exponent part of the term is NOT the same. For example, 2 3 5 !
√ √ √ or √ Radicals with different bases CANNOT be added together because they are NOT like terms!
So, for example, the following ways of rewriting radical expressions are correct:
√
√
√
4√3 2√3 2√3
√2√2 √2 ⋅ 2 And the following rewriting attempts are ALL incorrect!
2 √ 2
√1 √3 √1 3
4√3 2√3 6√9
4√3 2√3 6√6
2√2 4 3 6√2 ⋅ 3
2√2 4 3 6√2 3
Now you try! Use the allowed rules to rewrite each of the following expressions. If you cannot rewrite the
expression using one of the rules above, write CANNOT REWRITE as your answer:
√ √
4
√
√
√5 √11
5√5 √5 2√5
Factoring
In order to simplify radicals, we are going to need a systematic way to identify how the base under a square root sign
can be broken up into a product of squares. In order to do this, we will need to be able to factor numbers completely
into their simplest parts. Factoring just means that we rewrite a number as a product of smaller whole numbers. So,
for example, 6 could be factored as 6 2 ⋅ 3. There is often more than one way to factor a number. For example, 12
can be factored several different ways: 12 3 ⋅ 4 or 12 2 ⋅ 6. When we factor numbers completely, we break them
down into the smallest possible pieces, by continuing to factor every number until we can’t go any farther. If we do this
with 12 for example, we get: 12 2 ⋅ 2 ⋅ 3. Here are some more examples of numbers that have been completely factored:
20 2 ⋅ 2 ⋅ 5 100 2 ⋅ 2 ⋅ 5 ⋅ 5 63 3 ⋅ 3 ⋅ 7 300 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 5
Now you try! Factor each of the following numbers COMPLETETLY:
28
36
180
45
Finding squares once numbers have been factored completely
Once we have factored a number completely, we want to be able to identify squares in that factoring, and to rewrite
the product of the factor with as many squares as possible. So, for example, here is how we could rewrite the factoring
that we did above by first grouping factors together to make a square, and then rewriting them as a square:
20 2 ⋅ 2 ⋅ 5 2 ⋅ 2 ⋅ 5 2 ⋅ 5 100 2 ⋅ 2 ⋅ 5 ⋅ 5 2 ⋅ 2 ⋅ 5 ⋅ 5 2 ⋅ 5
63 3 ⋅ 3 ⋅ 7 3 ⋅ 3 ⋅ 7 3 ⋅ 7 300 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 5 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 5 2 ⋅ 3 ⋅ 5
Now you try! For each of the factorings that you did in the last section, (28, 36, 180 and 45), recopy those factorings
here, use parentheses to group together factors as needed to make as many squares as possible, and then rewrite
each grouping as a square. Use the examples just above as a reference about how to do this:
28
36
180
45
Rewriting radical expressions
Now that we understand the basic definitions and rules for what we can do with radicals, we want to apply these rules
to rewriting radical expressions, usually with the goal of simplifying. The basic rule for simplifying radicals is this:
If we can rewrite what is UNDER the square root sign as a list of SQUARES that are being either MULTIPLIED or
DIVIDED (NOT added or subtracted), then we can simplify the parts under the radical that are written as squares.
For example:
√28 √2 ⋅ 2 ⋅ 7 2 ⋅ 2 ⋅ 7 √2 ⋅ 7 √2 ⋅ √7 2√7
2√80 2√2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 2 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 2√2 ⋅ 2 ⋅ 5 2√2 ⋅ √2 ⋅ √5 2 ⋅ 2 ⋅ 2 ⋅ √5 8√5
√99 √3 ⋅ 3 ⋅ 11 3 ⋅ 3 ⋅ 11 √3 ⋅ 11 3√11
√28 √63 √2 ⋅ 2 ⋅ 7 √3 ⋅ 3 ⋅ 7
2 ⋅ 2 ⋅ 7 3 ⋅ 3 ⋅ 7
√2 ⋅ 7 √3 ⋅ 7
√2 √7 √3 √7
2√7 3√7
5√7
5√75 √12 2√27 5√3 ⋅ 5 ⋅ 5 1√2 ⋅ 2 ⋅ 3 2√3 ⋅ 3 ⋅ 3
5 3 ⋅ 5 ⋅ 5 1 2 ⋅ 2 ⋅ 3 2 3 ⋅ 3 ⋅ 3
5√3 ⋅ 5 1√2 ⋅ 3 2√3 ⋅ 3 5√3√5 1√2 √3 2√3 √3
5 ⋅ √3 ⋅ 5 1 ⋅ 2 ⋅ √3 2 ⋅ 3 ⋅ √3
25√3 2√3 6√3
25√3 6√3 2√3
31√3 2√3
29√3
3 5√3 5 2√12 3 5 5√3 2√12
3 5 5√3 2√2 ⋅ 2 ⋅ 3
8 5√3 2 2 ⋅ 2 ⋅ 3
8 5√3 2√2 ⋅ 3
8 5√3 2√2 √3
8 5√3 2 ⋅ 2√3
8 5√3 4√3
8 9√3
√ √
√
⋅√12 √2 ⋅ 2 ⋅ 3 2 ⋅ 2 ⋅ 3 √2 ⋅ 3 √2 √3 2√3
√5 √5 √15 √5√5 √5√15
√5 ⋅ 5 √5 ⋅ 15
√5 ⋅ 5 √5 ⋅ 3 ⋅ 5
√5 ⋅ 5 √3 ⋅ 5 ⋅ 5
5 ⋅ 5 3 ⋅ 5 ⋅ 5
√5 √3 ⋅ 5
√5 √3√5
5 √3 ⋅ 5
5 5√3
√2 1 3√2 4 √2 ⋅ 3√2 √2 ⋅ 4 1 ⋅ 3√2 1 ⋅ 4
3√2√2 4√2 3√2 4
3√2 ⋅ 2 1√2 4
3√2 1√2 4
3 ⋅ 2 1√2 4
6 1√2 4
6 4 1√2
2 √2
Now you try! Simplify the following radical expressions, or if they cannot be simplified, write CANNOT BE
SIMPLIFIED: (If you need more space, feel free to do all the work on your own paper!)
√45
2√27
√52
√45 √20
3√20 √45 2√80
8 2√5 9 7√20
√ √
√
√3 √6 √3
√3 2 2√3 1
