ALGEBRA 2
Name:_________________________________ Date:_____________
Multiplying and Dividing nth Roots:

If multiplying or dividing radicals with nth roots, use the same properties as with square roots.
𝑛
𝑛
𝑛
o Product Property: √𝑎𝑏 = √𝑎 ∙ √𝑏
𝑛
𝑎
𝑛
𝑏
𝑛
o Quotient Property: √ =
√𝑎
√𝑏
Examples:
3
3
6
1) √3 ∙ √9 =
6
2) √8 ∙ √8 =
3
3)
√128
3
√2
=
4
4)
√48
4
√3
=
Simplifying nth Roots:


Method 1: Make a factor tree, match the common factors and put the uncommon factors back under the
radical sign. (For a match, you need as many of the same prime factors as the root.)
- For example, if the root was a 3, you would need “3 of a kind” to have a “match”.
Method 2: Find the largest perfect nth root that divides evenly into the radicand. Use the Product Property
to simplify.
o Perfect Squares: ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, …
o
Perfect Cubes: ____, ____, ____, ____, ____, ____, ____, ____, ____, ______, …
o
Perfect Fourths: ____, ____, ____, ____, ____, ______, ______, …
o
Perfect Fifths: ____, ____, ____, ______, ______, …
Examples: Write each expression in simplest form. No decimal answers!!
Let’s start with what you already know: √12
3
5) √40 =
3
6) √128 =
4
7) √243 =
5
8) √128 =
Remember:

Radicals are another way of writing fractional exponents… SO, all of your exponent rules may also be applied to
roots!
Properties of Rational Exponents:

Same properties as with integer exponents
Bases stay
the same!
Examples: Simplify each expression.
2
3
1
3
9) 6 ∙ 6 =
3
4
4
1
2
10) (3 ) =
11) (16 ∙ 25) =
Radicals must be the same
Add/Subtract only the coefficients
Radicals stay the same
16) 5√6 + 8√6 =
5
5
17) 4√3 − √3 =
3
3
1
6
1
3
24) 9 ∙ 9 =
2
3 3
4
28) (25 ) =
5
5
21) √4 ∙ √8 =
1
1
3
22)
√54
3
√2
5
25)
818
3
818
29) 16
−
1
4
6
=
13)
72
1
72
=
2
2
19) 9 (35 ) − 12 (35 ) =
23)
1
=
5
3
=
26) (64 ∙ 49)2
=
1
3
15) 4𝑥 2 − 8𝑥 2 =
18) 7 (113 ) + 10 (113 ) =
Practice:
20) √2 ∙ √4 =
12) 8
Review: Simplify.
14) 5𝑥 + 2𝑥 =
Adding and Subtracting Radicals:



−
6
30) 5√2 + 6 √2 =
√192
3
√3
=
1
1
27) 3 (22 ) + 5 (22 )
1
1
31) (72 ) − 9 (72 ) =
Simplifying Expressions with Variables: Write your answer using positive exponents only. Assume all
variables are positive.
32)
√9𝑥 6
=
3
33) √
𝑥3
𝑦6
=
3
1
34) (4𝑦 6 )2 =
35)
3𝑎2 𝑐
𝑎𝑐 −2
=
Practice: Simplify the expression. Write your answer using positive exponents only. Assume all variables are
positive.
36) √25𝑦 4 =
𝑥6
37) √
𝑦2
=
1
38) (8𝑢3 𝑣 9 )3 =
39)
2𝑥
1
𝑥 3 𝑧 −3
Classwork/Homework: Text page 362-363: #20-26 even, 28-30, 36-40, 42-49, 50-64 even, 66-67
=