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```MAT 086 – Intermediate Algebra-Part II
MAT 095 – Intermediate Algebra
Section 10.3 (Bittinger CA6)
Page 1
Topics to be Covered and Student Learning Outcomes (SLOs)
After we cover Section 10.3, you should be able to:
1. State the Product Rule for Radicals, and use the Product Rule to:
a. Multiply radicals with the same index. [SLO 4]
b. Simplify radicals, by factoring out perfect n-th powers from the radicand.
[SLO 4]
2. Recognize expressions with variables that are perfect n-th powers. [SLO 4]
Properties of Exponents and Properties of Radical Expressions
In Sections 10.1 and 10.2 we have learned that:


Radical expressions can be written as exponential expressions with rational
(fractional) exponents, and vice versa, and
We can simplify a radical expression as follows:
o Convert the radical expression to an exponential expression with
rational exponents.
o Use arithmetic and the properties of exponents to simplify the
exponential expression.
o Convert the simplified exponential expression back to radical notation,
as needed.
In Sections 10.3 and 10.4, we will use the properties of exponents (from Section 4.1) to
derive two properties or “rules” that can be used to simplify and perform arithmetic
expression.
Product Rule for
(based
on the Power of a Product Rule for Exponents)
The product of two n-th roots is the n-th root of the
n a n b  n ab
where
na
and
nb
are real numbers.
The Product Rule for Radicals can be used to multiply
NOTE: Be careful
the Product
Rule!
twonot
(ortomore)
with
the same index.
 The Product Rule for Radicals is used to multiply radicals with the same index.

For example: n a  m b  nm a  b
MAT 086/095 (Bittinger CA6) – 10.3-10.4
Slide 4
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MAT 086 – Intermediate Algebra-Part II
MAT 095 – Intermediate Algebra
Section 10.3 (Bittinger CA6)
Page 2
The Product Rule for Radicals is derived from the Power of a Product Rule for
exponents:
n a  b  a  b 1/n


Convert the radical expression to a rational exponent.
 a1/n  b1/n
Apply the Power of a Product Rule.
 na n b
Convert the rational exponent to a radical.
Multiply:
1.
4  25
2.
5 6
3.
3
7 3 9
4.
4
x
6
4
3
y
5.
x 3  x 3
2
1/2
NOTE: We’ve already seen that  a  b   a2  b2 . Similarly  a  b 
 a1/2  b1/2 .
Rewriting this in radical notation, we get:
a2  b2  a2  b2
For example:
x 2  3  x2  32
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MAT 086 – Intermediate Algebra-Part II
MAT 095 – Intermediate Algebra
Section 10.3 (Bittinger CA6)
Page 3
We generally write numbers in the simplest form. For example, we generally reduce
fractions, writing 4 as 1 .
8
2
Similarly, the square root of a perfect square or the cube root of a perfect cube (or the
n-th root of a perfect n-th power) can be rewritten as an integer. For example:
16  4
3
27  3
In other words:
 We would never leave a perfect square under a square root sign, and
 We would never leave a perfect cube under a cube root sign, and
 We would never leave a perfect n-th power under an n-th root sign.
Radical expressions with radicands that are not perfect n-th powers can also be
simplified by factoring out a perfect n-th power from the radicand.
EXAMPLE: Simplifying Square Roots by Factoring
Square roots of expressions that are not perfect squares can be simplified by factoring
out a perfect square from the radicand, for example:
20  4  5
Factor out 4 from the radicand, because 4 is the largest
perfect square that is a factor of 20.
 4 5
According to the Product Rule for Radicals.
2 5
Evaluating
4
Note that 20 is not considered to be simplified because there is a perfect square factor
(namely, 4) under the square root sign!
EXAMPLE: Simplifying Cube Roots by Factoring
Cube roots of expressions that are not perfect cubes can be by factoring out a perfect
cube from the radicand, for example:
3
24  3 8  3
 3 8 33
 
2 33
Factor out 8 from the radicand, because 8 is the largest
perfect cube that is a factor of 24.
According to the Product Rule for Radicals.
Evaluating 3 8
Note that 3 24 is not considered to be simplified because there is a perfect cube factor
(namely 8) under the cube root sign!
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MAT 086 – Intermediate Algebra-Part II
MAT 095 – Intermediate Algebra
Section 10.3 (Bittinger CA6)
Page 4
In other words, to simplify a radical expression with index n, factor out any perfect n-th
 Write the radicand as a product in which one factor is the largest perfect n-th
power.
 Using the Product Rule for Radicals, take the n-th root of each factor.
 Simplification is complete when no radicand has a factor that is a perfect n-th
power.
NOTE: In order to simplify radicals, you must be able to recognize perfect squares,
perfect cubes, etc.
Simplify each of the following radical expressions.
From now on, we will assume that a radicand does not represent a negative number raised
to an even power. Thus, we will not need to use absolute value notation when finding even
roots.
6.
80
7.
18m10n7
8.
3
54s8
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MAT 086 – Intermediate Algebra-Part II
MAT 095 – Intermediate Algebra
Section 10.3 (Bittinger CA6)
Page 5
Recognizing Perfect n-th Powers
Any exponent that is a multiple of n represents a perfect n-th power.
Perfect squares
(Exponent is a multiple of 2)
2 4 6 8
x , x , x , x ,...
NOTE: 9 is a perfect square,
but x 9 is not a perfect square!
Perfect cubes
(Exponent is a multiple of 3)
3 6 9 12
x ,x ,x ,x
,...
NOTE: 8 is a perfect cube,
but x 8 is not a perfect cube!
Perfect 4th powers
(Exponent is a multiple of 4)
4 8 12 16
x ,x ,x
,x
To find the n-th root, divide the
exponent by n:
   x12
1/3
3 24
x   x 24 
 x8
x 24  x 24
1/2
,...
NOTE: 81 is a perfect 4th power,
but x 81 is not a perfect 4th power!
Simplify each of the following radical expressions.
9.
x 9 y25z 49
10. 3 16x10 y6
11. 4 16x 9 y7
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MAT 086 – Intermediate Algebra-Part II
MAT 095 – Intermediate Algebra
Section 10.3 (Bittinger CA6)
Page 6
Use the Product Rule for Radicals to multiply, then simplify as necessary.
12.
10  6
13. 3 9x3 y2  3 9x 4 y7
14.
15x3 y2  5 y 4
15. 3 2x 2 y 4  3 4x 2 y6