Product Rule for Radicals:
- when indices are the same, radicands can be multiplied if all the roots
exist
- this can be used to combine radicals or break them apart
o √3 ∙ √7 = √6 ∙ 5
○ √27 = √25 ∙ 2
= √30
= √25 ∙ √2
= 5√ 2
- in some instances we will need to use the Product Rule to do both;
combine two radicals first and then break them apart
3
3
o √6 ∙ √15 = √6 ∙ 5
○ √4 ∙ √6 = √6 ∙ 5
Quotient Rule for Radicals:
- when indices are the same, radicands can be divided, if all the roots
exist
- this can be used to combine radicals or break them apart
o
√35
√7
15
= √3
= √5
2
○ √9 =
√3
√16
=
√3
4
Simplifying Radicals:
- remove factors from the radical until no factor in the radicand has a
degree greater than or equal to the index
o √𝑥 5 is not simplified because the degree of the radicand (5) is
larger than the index (2)
- use the Product (or Quotient) Rule for Radicals to simplify (as shown
below on the left), or use fractional exponents (as shown below on
the right)
o
√𝑥 5
= √𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥
= √ 𝑥 ∙ √𝑥 ∙ √𝑥 ∙ √𝑥 ∙ √𝑥
= 𝑥 ∙ 𝑥 ∙ √𝑥
2
= 𝑥 ∙ √𝑥
= 𝑥 2 ∙ √𝑥
○
√𝑥 5
=𝑥
=𝑥
=𝑥
5
2
1
2
2
2+
2
1
2
1
2
=𝑥 ∙𝑥
= 𝑥 2 ∙ √𝑥
Example 1: Simplify the following radical expressions completely. If you
convert to fractional exponents, be sure to reduce those exponents as well
(no improper fractions). If you convert to fractional exponents, be sure to
reduce those exponents as wel (assume that all variables are positive).
a. √9𝑥 2 𝑦 4
b. √32𝑎8 𝑏5
b.
3
3
c. √64𝑥 9 𝑦 6
d.
d. √16𝑥 3 𝑦 −8 𝑧 4
e. √2𝑥 3 𝑦 5 √8𝑥𝑦
f.
f. √15𝑥𝑦 −7 √10𝑥 −3 𝑦 3
3
3
g. √3𝑡 4 𝑣 2 ∙ √−9𝑡𝑣 5
3
3
h. √25𝑎−4 𝑏7 ∙ √10𝑎 9 𝑏𝑐 10
We have seen the Product Rule for Radicals and the Quotient Rule for
Radicals, and we have seen how to use them to simplify radicals.
However, there is no Sum Rule for Radicals or Difference Rule for
Radicals:
-
√𝑥 2
+
𝑦2
1
2
2 )2
2
2 )2
= (𝑥 + 𝑦
1
,
√𝑥 2 + 𝑦 2 ≠ 𝑥 + 𝑦
BUT
−
= (𝑎 − 𝑏 ,
BUT
√𝑎2 − 𝑏2 ≠ 𝑎 − 𝑏
- when a sum or difference is raised to a power, the power is NOT
distributive
o √32 + 42 ≠ 3 + 4
○ √52 − 42 ≠ 5 − 4
√𝑎2
𝑏2
√9 + 16 ≠ 7
√25 − 16 ≠ 1
√25 ≠ 7
√9 ≠ 1
5≠7
3≠1
Example 2: Simplify the following expressions completely (assume that
all variables are positive).
3
2
a. (4 + 𝑥 )
b. 𝑎
3
2
b. 4 + 𝑥
3
2
5
2
d. 9 − 𝑥
3
2
5
3
c. (9 − 𝑥 )
d. 𝐹
e. (8 + 𝑥 )
5
2
f. 8 + 𝑥
Answers to Examples:
1a. 3𝑥𝑦 2 ; 1b. 4𝑎4 𝑏2 ∙ √2𝑏 ; 1c. 4𝑥 3 𝑦 2 ; 1d.
1e. 4𝑥 2 𝑦 3 ; 1f.
5√6
𝑥𝑦 2
; 1g.
3
2𝑥𝑧
𝑦2
3
2𝑧
∙ √𝑦 2 ;
3
−3𝑡𝑣 2 √𝑡 2 𝑣 ; 1h. 5𝑎𝑏2 𝑐 3 √2𝑎2 𝑏2 𝑐 ;
2b. 4 + 𝑥√𝑥 ; 2f. 32 + 𝑥 ; 2d. 9 − 𝑥 2 √𝑥 ; 2a. (4 + 𝑥 ) √4 + 𝑥 ;
2c. (9 − 𝑥 )2 √9 − 𝑥 ; 2e. (8 + 𝑥) √8 + 𝑥 ;