The document defines key terms related to radical expressions, including radical expressions, radicands, and simplest form. It then outlines several properties of radicals: the product property, which allows multiplying radicals by splitting them into factors; the quotient property, which allows simplifying radicals in fractions; and rationalizing denominators, which removes radicals from denominators. Finally, it provides examples applying these properties to simplify, add, subtract, and multiply radical expressions.

2. Vocabulary
 Radical Expression – expression that contains a
radical, such as a square root, cube root, or other root.
 Simplest form of a radical expression:
 No perfect square factors are in the radicand.
 No fractions are in the radicand.
 No radicands appear in the denominator of a fraction.

3. Product Property of Radicals
 The square root of a product equals the product of the
square roots of the factors.
ab = a × b where a ³ 0 and b ³ 0
4x = 4 × x = 2 x

4. Example 1 Use the product property of radicals
=
a. 32 16 • 2 Factor using perfect square factor.
=
16 • 2 Product property of radicals
=
4 2 Simplify.
9 • x2 • x
=
b. 9x3 Factor using perfect square factors.
9 • x2 •
=
x Product property of radicals
=
3x x Simplify.

5. Example 2 Multiply radicals
2• 8 2•8
=
a. Product property of radicals
=
16 Multiply.
=
4 Simplify.
3x • 4 x 4 3x • x
=
b. Product property of radicals
=
4 3x2 Multiply.
4• 3•
=
x2 Product property of radicals
=
4x 3 Simplify.

6. Example 2 Multiply radicals
7xy2 • 3 x 3 7xy2 • x
=
c. Product property of radicals
=
3 7x2y2 Multiply.
3• 7• x2 • y2 Product property of radicals
=
=
3xy 7 Simplify.

7. Quotient Property of Radicals
 The square root of a quotient equals the quotient of
the square roots of the numerator and denominator.
a a
= where a ³ 0 and b > 0
b b
16 16 4
= =
25 25 5

8. Example 3 Use the quotient property of radicals
a. 13 13 Quotient property of radicals
=
100 100
13
= Simplify.
10
b. 7 7
2
= Quotient property of radicals
x x2
7 Simplify.
=
x

9. Rationalizing the denominator
 The process of eliminating a radical from an
expression’s denominator is called rationalizing the
denominator.

10. Example 4 Rationalize the denominator
5 5 7 7
a. = • Multiply by .
7 7 7 7
5 7
= Product property of radicals
49
5 7
= Simplify.
7
2 2 3b 3b
b. = • Multiply by .
3b 3b 3b 3b

11. Example 4 Rationalize the denominator
6b
= Product property of radicals
9b2
6b
= Product property of radicals
9 • b2
6b
= Simplify.
3b

12. Example 5 Add and subtract radicals
Commutative
a. 4 10 + 13 – 9 10 = 4 10 – 9 10 + 13 property
Distributive
= (4 – 9) 10 + 13 property
= – 5 10 + 13 Simplify.
Factor using perfect square
b. 5 3 + 48 = 5 3 + 16 • 3 factor.
= 5 3 + 16 • 3 Product property of radicals
= 5 3 +4 3 Simplify.

13. Example 5 Add and subtract radicals
= (5 + 4) 3 Distributive property
= 9 3 Simplify.

14. Example 6 Multiply radical expressions
a. 5 (4 – 20 ) = 4 5 – 5 • 20 Distributive property
= 4 5 – 100 Product property of
radicals
= 4 5 – 10 Simplify.
b. ( 7 + 2 )( 7 – 3 2 )
= 7( 7 – 3 2) + 2( 7 – 3 2 ) Distributive
property
2
= ( 7 ) + 7 (– 3 2) + 2 • 7 + 2 (– 3 2) Distributive
property

15. Example 6 Multiply radical expressions
= 7 – 3 7 • 2 + 7 • 2 – 3 ( 2 )2 Product property
of radicals
= 7 – 3 14 + 14 – 6 Simplify.
= 1 – 2 14 Simplify.

16. 8.4 Warm-Up (Day 1)
1. 48
2. 4x 2
3.
50 × 18
4.
3b × 18b
3
5. 7
81

17. 8.4 Warm-Up (Day 2)
1. 75xy × 2x 3
4
2.
3
3. 1
2x
4.
2 6 - 5 54
5.
(
6 7 3+6 )