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Haberman MTH 95
Section IV: Radical Expressions, Equations, and Functions
Module 3: Multiplying Radical Expressions
Recall the property of exponents that states that. We can use this rule to obtain an analogous rule for radicals:
a bm^ m = ( ab ) m
1 1 ( )^1 (using the property of exponents given above)
n n^ n^ n n n
a b a b ab ab
Product Rule for Radicals If a and b are positive real numbers and n is a positive integer, then n^ a โ
n^ b = nab.
EXAMPLE: Perform the indicated multiplication, and simplify completely. a. 2 โ
18 b.^4 3 x^2 โ
^427 x^2
SOLUTIONS:
2
(product rule for radicals)
(write as a perfect square)
a.
4 2 4 2 4 2 2 (^4 2 ) 4 4 4 4 4
(product rule for radicals)
(product rule for radicals) (we need to use the absolute value since is even)
x x x x x x x x x
b.
Product Rule for Simplifying Radical Expressions: When simplifying a radical expression it is often necessary to use the followingequation which is equivalent to the product rule: n (^) ab = n (^) a โ
nb.
EXAMPLE: Simplify 40.
SOLUTION: Since 40 isnโt a perfect square, we need to write 40 as a product containing a factor that is a perfect square:
40 4 10 4 2 1
(factor using perfect square(s)) (product rule for simplifying radical expressions)
EXAMPLE: Simplify the following. a.^3 24 b.^4 16 w^8 c. 54 d^5
SOLUTIONS: 3 3 3 3 3
(factor 24 using perfect cube(s)) (product rule for simplifying radical expressions)
a.
( )
(^4 8 ) 4 4 4 2 4 2 2
(product rule for simplifying radical expressions)
(we don't need the absolute value here since must be positive)
w w w w w
b.
Try these yourself and check your answers. Perform the indicated multiplication, and simplify completely. a. t โ
8 t^3 b.^3 2 p^2 โ
3 p
SOLUTIONS: (^8 3) 1 2 3 8 1 2 3 8 4 8 3 8 7 8 (^8 )
(write with rational exponents) (use a property of exponents) (create a common denominator for the exponents) (use another property of exponents) (write final answ
t t t t t t t t
a.
er in radical notation to agree with the original expression)
( ) (^ )
( ) (^ )
( ) ( (^ ))
3 2 2 1 3^ 1 2 2 2 6^ 3 6 2 2 1 6^3 1 6 4 3 1 6 7 1 6 6 7 6 6 6
(write with rational exponents) (create a common denominator for the exponents)
p p p p p p p p p p p p p p p p
= โโ^ โโ
b.
