Algebra 2
11.2 Notes
11.2(Day 1) Simplifying Radical Expressions
Date: __________
Establishing the Properties of Rational Exponents
In the past, you have used properties of integer exponents to simplify and evaluate expressions. These
properties are the same for rational exponents. Recall the properties below:
Properties of Rational Exponents
For all nonzero real numbers a and b and rational numbers m and n.
Words
Product of Powers
Property
To multiply powers with the
same base, add the exponents.
Quotient of Powers
Property
To divide powers with the
same base, subtract the
exponents.
Power of a Power
Property
To raise one power to another,
multiply the exponents.
Algebra
Numbers
1
𝑎𝑚 ∙ 𝑎𝑛 = _______________
2
𝑎𝑚
𝑎𝑛
1253
= _______________
2
(𝑎𝑚 )𝑛
Power of a Product
Property
To find a power of a product,
distribute the exponent.
Power of a Quotient
Property
To find the power of a
quotient, distribute the
exponent.
=
1
1253
3
(83 ) =
= ______________
Negative Power Property
When a number is raised to a
negative power, it is equal to
its reciprocal raised to the
positive (opposite) power.
3
122 ∙ 122 =
1
1 𝑚
𝑎 −𝑚
𝑎−𝑚 = (𝑎) or (𝑏)
𝑏 𝑚
= (𝑎)
16−2 =
5
1 −3
(64) =
1
(𝑎𝑏)𝑚 = _____________
(16 ∙ 25)2 =
1
𝑎 𝑚
(𝑏) = ____________
16 4
(81)
=
1
Algebra 2
11.2 Notes
Recall the Rational Exponent Property from 11.1:
𝑚
𝑛
𝑛
𝑎 𝑛 = √𝑎 𝑚 = ( √𝑎 )
𝑚
Rewrite each expression with rational exponent as a radical expression and simplify.
1) 27
2
3
2) 64
1
5
2
3)
9 2
(49)
Simplifying Rational-Exponent Expressions
Learning Target B: I can use the properties of rational exponents to simplify rational-exponent
expressions.
Simplify the expression. Assume that all variables are positive. Exponents in simplified form
should all be positive.
3
5
A) 25 ∙ 25
C) (
1
7
5
B)
83
2
83
3
2
4
𝑦3
2
16𝑦 3
2
3
2
3 3
4
)
D) (27𝑥 )
3
4 2
3
E) (12 ∙ 12 )
1 2
F)
(6𝑥 3 )
5
𝑥3𝑦
2
Algebra 2
11.2 Notes
Simplifying Radical Expressions Using the Properties of Exponents
Learning Target C: I can use the properties of exponents to simplify radical expressions.
Simplify the expression by writing it using rational exponents and then using the properties of
rational exponents. Assume that all variables are positive. Exponents in simplified form
should all be positive.
3
A) 𝑥( 3√2𝑦)( √4𝑥 2 𝑦 2 )
√𝑥 3
C) 3
√𝑥 2
E)
2√𝑥 5 𝑦 2
6
8 √𝑥 15 𝑦
√64𝑦
B) 3
√64𝑦
6
4
3
D) √163 ∙ √46 ∙ √82
3
F) 𝑦 3 ( √𝑥𝑦 5 )
3
Algebra 2
11.2 Notes
11.2 (Day 2) Simplifying Radical Expressions
Date: __________
Simplifying Radical Expressions Using the Properties of 𝒏𝒕𝒉 Roots
Learning Target D: I can use the properties of 𝑛𝑡ℎ to simplify radical expressions.
From working with square roots, you know, for example that
√8 ∙ √2 = ______ = ________
√8
and
√2
= ______ = _______
The corresponding properties also apply to 𝑛𝑡ℎ roots.
Properties of 𝒏𝒕𝒉 Roots
For 𝑎 > 0 and 𝑏 > 0
Words
Product Property of Roots
The 𝑛𝑡ℎ root of a product is
equal to the product of the 𝑛𝑡ℎ
roots.
Algebra
Numbers
𝑛
𝑛
Quotient Property of Roots
The 𝑛𝑡ℎ root of a Quotient is
equal to the Quotient of the 𝑛𝑡ℎ
roots.
3
√𝑎𝑏 = 𝑛√𝑎 ∙ √𝑏
𝑛
𝑎 √𝑎
√ =𝑛
𝑏
√𝑏
𝑛
3
3
√16 = √8 ∙ √2 = _____________
25
√ =
16
Simplify the expression using the properties of 𝒏𝒕𝒉 roots. Assume that all variables are
positive.
4
81
3
B) √ 8
C) √54𝑥12 𝑦 15
4
D) √𝑥 16
E) √75𝑥 7 𝑦 8 𝑧
F)
A) √27𝑥 3 𝑦 9
4
24
3
3
√2∙ √12
3
√3
4