Download U7D2 Notes Rational Exponents

Rational Exponents
N.RN.2 – Rewrite expressions involving radicals and rational
exponents using properties of exponents.
N.RN.3 – Explain why the sum or product of two rational
numbers is rational; that the sum of a rational number and an
irrational number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational.
How to play…
• When a new problem is shown, write down your
answer on a slip of paper (write it BIG)
• Do not show your answer to anyone!
• When the teacher says “SHOWDOWN” slap your
answer down for everyone to see.
• Discuss your answers with your group, come to
an agreement on the correct answer
Showdown
−24
(-3)
4
169
−2
8
2
2
5 ∙5
53
3
𝑥2
24 ∙ 3−2
2𝑥
2
= 21 … what is x?
Rational:
𝟏
𝟑
, −𝟐, 𝟎. 𝟏, − …
𝟑
𝟕
Irrational:
𝝅 ≈ 𝟑. 𝟏𝟒𝟏𝟓 …
𝟐 ≈ 𝟏. 𝟒𝟏𝟒𝟐 …
Integer:
… − 𝟐, −𝟏, 𝟎, 𝟏, 𝟐 …
Whole:
𝟎, 𝟏, 𝟐, 𝟑 …
Natural:
𝟏, 𝟐, 𝟑 …
Sets of Real Numbers
Always, Sometimes, Never
1.
2.
3.
4.
The sum of two rational numbers is rational
The product of two rational numbers is a
whole number
The sum of a rational number and an irrational
number is rational
The product of a nonzero rational number and
an irrational number is irrational
Always, Sometimes, Never
1.
2.
3.
4.
The sum of two rational numbers is rational (A)
The product of two rational numbers is a
whole number (S)
The sum of a rational number and an irrational
number is rational (N)
The product of a nonzero rational number and
an irrational number is irrational (A)
Properties of Exponents
Name
Property
Example
Product of Powers
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
Quotient of Powers
𝑎𝑚
𝑚−𝑛
=
𝑎
𝑎𝑛
Power of a Product
Power of a Quotient
Power of a Power
Negative Exponent
𝑎∙𝑏
𝑛
= 𝑎𝑛 ∙ 𝑏𝑛
𝑎
𝑏
𝑛
𝑎𝑚
𝑛
𝑎−𝑛
23 ∙ 22 =
25
23
=
2∙3
2
𝑎𝑛
= 𝑛
𝑏
2 2
3
=
= 𝑎𝑚𝑛
23
=
1
= 𝑛
𝑎
2
2−3 =
=
The “nth” root of a
Index
𝑛
𝑎
Radicand
Properties of Radicals
Name
Radical of a Product
Radical of a
Quotient
Radical of a Radical
Property
𝑛
𝑎𝑏 =
𝑛
𝑛 𝑚
Example
𝑛
𝑎
=
𝑏
𝑎=
𝑎∙
𝑛
𝑛
𝑛
𝑏
𝑎
3
9∙ 3=
75
3
𝑏
𝑛𝑚
3
3
𝑎
=
64 =
Rational Exponents & Radicals
Exponent
Radical
1
𝑎𝑛
𝑚
𝑎𝑛
=
𝑛
1 𝑚
𝑎𝑛
Example 1.
3
𝑛
3
𝑎m
5 =
Example
1
2
𝑎
or
9 =
𝑛
𝑎
𝑚
3
2
4 =
Example 2. 5
1
3
3
=
With your partner…
3
1
3
Explain why it makes sense that 𝑎 = 𝑎 and
3
𝑎2 = 𝑎
2
3
Example 3: Simplify each expression. Express
solutions in radical form (where necessary).
A)
3
𝑥𝑦
6
=
B) 𝑥 ∙ 3 𝑥 =
C)
𝑥
4
𝑥
=
Example 4.
In parts B & C, you started with an expression in
radical form, converted to rational exponent
form, and then converted back to radical form.
Explain the purpose of each conversion.
Example 5.
A) 27𝑥 9
2
3
B) What is the simplified form of 27𝑥 9
2
3
How is it related to 27𝑥 9 ?
2
−
3
?
Example 6.
2
5
2
5
1
3
−
1 3
78 ∙78
5
78
=
A) 4 ∙ 4 =
B) 5 ∙ 5
C)
4
3
=
Example 6.
D) 2k 𝑘
E) 𝑥
F)
6
5
1
3 8
8
3
5
−
𝑥 𝑦
1 4
2𝑦 5
3
𝑦 10
=
=
1
1 3
4
=
Example 6.
2𝑥 2 + 8𝑥 2 =
G)
H)
3
54𝑥 4 𝑦
3
− 𝑥 2𝑥𝑦 =
Exit Card
1)
A) 49 =
3
C) 8 =
2)
−49 =
B)
D)
3
−8 =
What are rational and irrational numbers and
how are radicals related to rational exponents?