Download § 7.1 Radical Expressions and Radical Functions

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§ 7.1 Radical Expressions
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Definition of nth roots
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Definition of nth roots
What is the square root of a number? What is
the cube root of a number?
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Definition of nth roots
Definition: The square root of a number, x, is
the number that you multiply by its self two
times to get the number x. ( 𝑥)
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Definition of nth roots
Definition: The square root of a number, x, is
the number that you multiply by its self two
times to get the number x. ( 𝑥)
The cube root of a number, x, is the number
that you multiply by its self three times to get x.
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𝑥
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Definition of nth roots
Definition: The square root of a number, x, is
the number that you multiply by its self two
times to get the number x. ( 𝑥)
The cube root of a number, x, is the number
that you multiply by its self three times to get x.
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𝑥
The symbol,
is called a radical symbol.
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Definition of nth roots
Definition: The square root of a number, x, is
the number that you multiply by its self two
times to get the number x. ( 𝑥)
The cube root of a number, x, is the number
that you multiply by its self three times to get x.
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𝑥
The symbol,
is called a radical symbol.
An algebraic expression containing a radical is
called a radical expression.
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nth Roots
𝑎
3
𝑎
4
𝑎
5
𝑎
- Square root of a.
- Cube root of a.
- Forth root of a.
- Fifth root of a.
𝑛
- 𝑛𝑡ℎ root of a.
𝑎
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nth Roots
𝑎
3
𝑎
4
𝑎
5
𝑎
- Square root of a.
- Cube root of a.
- Forth root of a.
- Fifth root of a.
𝑛
- 𝑛𝑡ℎ root of a.
𝑎
Examples:
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Note:
When ever you have even roots ( 𝑥, 4 𝑥. 6 𝑥, …),
you can not have a negative under the roots.
The number would not be a real number, but an
imaginary number.
Though,…
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Note:
When ever you have even roots ( 𝑥, 4 𝑥. 6 𝑥, …),
you can not have a negative under the roots.
The number would not be a real number, but an
imaginary number.
Though,…
(−5)2 is O.K. since,
(−5)2 = 25 = 5
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Note:
When ever you have even roots ( 𝑥, 4 𝑥. 6 𝑥, …),
you can not have a negative under the roots.
The number would not be a real number, but an
imaginary number.
Though,…
(−5)2 is O.K. since,
(−5)2 = 25 = 5
Solutions to even roots should only be
positive/zero, unless, there is already a
negative outside the root.
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Properties of Roots
1.)
𝑛
𝑛
𝑥 𝑛 = 𝑥, if n is an odd positive integer.
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371 7 = 371
𝑥 𝑛 = 𝑥 , if n is an even positive integer.
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16𝑦 4 = 2 𝑦
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Properties of Roots
1.)
𝑛
𝑛
𝑥 𝑛 = 𝑥, if n is an odd positive integer.
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371 7 = 371
𝑥 𝑛 = 𝑥 , if n is an even positive integer.
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16𝑦 4 = 2 𝑦
𝑛
𝑛
2.) 𝐴 ∙ 𝐵 = 𝐴 ∙
or expressions.
225𝑦 6 =
𝑛
𝐵, where A and B are numbers
9 ∙ 25 ∙ 𝑦 2 ∙ 𝑦 2 ∙ 𝑦 2 = 9 25 𝑦 2 𝑦 2 𝑦 2
=3∙5∙ 𝑦 ∙ 𝑦 ∙ 𝑦
= 15 𝑦 3
𝑛
𝑛
2.) 𝐴 ∙ 𝐵 = 𝐴 ∙
or expressions.
225𝑦 6 =
3.)
𝑛
𝐴
𝐵
=
𝑛
𝐵, where A and B are numbers
9 ∙ 25 ∙ 𝑦 2 ∙ 𝑦 2 ∙ 𝑦 2 = 9 25 𝑦 2 𝑦 2 𝑦 2
=3∙5∙ 𝑦 ∙ 𝑦 ∙ 𝑦
= 15 𝑦 3
𝑛
𝐴
𝑛
𝐵
;
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125
−64
3
=
3
125
−64
=
5
−4
=
5
−
4
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