MAT 51 Wladis Project 5: Simplifying Expressions of all types Use the following properties to simplify the following expressions. 1 1 ⋯
0 Multiple instances of addition can be performed in any order. For example: Multiple instances of multiplication can be performed in any order. For example: ⋯
⋯
⋅ …⋅
⋯
⋯
0 1
0 ⋅ ,
√
√
0 ⋅…⋅
0 ⋯
0 √ √ √
√
0 ⋅…⋅
⋅ …⋅
⋯
,…,
0 0 Simplify as much as possible, using ONLY the identities given in this project: While simplifying, rewrite all negative exponents as positive exponents. 1.
2.
3.
4.
5
3
3
3 1 2
7 1
2
7 5.
6.
2
7
7.
8.
9.
3√2 7√2
4
3 10.
11.
7
3
12.
13.
2 2
1
14.
15.
16.
√2 3 2
2
5
3
2 5
2
2 2√3
1 18.
√3
20.
21.
22.
1 17.
19.
3 2
3
2 2
3
3 4
3
2
3
5
7 2
For the following questions, factor the numbers under the radicals first—this will allow you to simplify further. For examples, see the next page. 23.
24.
25.
26.
27.
28.
√45 2√27 29.
√
30.
√3 √6
√52 √45 √20 3√20 √45
8 2√5 9
√
√
2√80 7√20 √3 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 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