Transcript Radical (Square Root) Expressions

Radical (Square Root)
Expressions
Definition of a Square Root
• The ancient Greeks knew to find the area of a square by multiplying
the length of its side by itself
• So, if a square has side length 8, its area is 8 ⋅ 8 = 82
• For this reason, today we say “8 squared” when we see 82
• If we wish to find the square of some number (like 7), we multiply it
by itself (we square the number): 7 ⋅ 7 = 72 = 49
• But what if, given some number, we are interested in finding the
number that, when squared, produces that number?
• For example, what if we wished to know what number is squared to
produce 121?
Definition of a Square Root
• We would like to have a symbol that means “this is the number that,
when squared, produces this value”
• The symbol that mathematicians have come to use is this one:
• This symbol is called a radical, or square root
• It works like this: suppose that 𝑦 = 𝑥, where 𝑥 ≥ 0. Then 𝑦 2 = 𝑥.
• For example, if 𝑦 = 121, then 𝑦 2 = 121
• This means that 𝑦 = 11 since 112 = 121
Definition of a Square Root
• We can also work this from the other side
• If 7 = 𝑥, then 72 = 𝑥 or 𝑥 = 49
• That is, 49 = 7
• In general
• If 𝑦 = 𝑥, then 𝑦 2 = 𝑥
• If 𝑦 2 = 𝑥, then 𝑦 = 𝑥 (this version is not complete, but we’ll see why later)
Guided Practice
Use the definition of square roots, 𝑦 = 𝑥 ⟺ 𝑦 2 = 𝑥, to rewrite the
following.
a) If 6 = 𝑥, then__________
b) If 𝑦 = 49, then __________
c) If 𝑦 = 2, then __________
d) If 24 = 576, then __________
e) If 225 = 15, then __________
f) If 12 = 144, then __________
Guided Practice
Use the definition of square roots, 𝑦 = 𝑥 ⟺ 𝑦 2 = 𝑥, to rewrite the
following.
a) If 6 = 𝑥, then 62 = 𝑥
b) If 𝑦 = 49, then 𝑦 2 = 49
c) If 𝑦 = 2, then 𝑦 2 = 2
d) If 24 = 576, then 242 = 576
e) If 225 = 15, then 225 = 152
f) If 12 = 144, then 122 = 144
Radicals and Exponents
How can we describe the pattern shown below?
•
•
•
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•
64 =
16 =
81 =
225 =
64 =
729 =
82 = 8
24 = 2 2 = 4
34 = 3 2 = 9
152 = 15
26 = 2 3 = 8
36 = 33 = 27
The pattern is _____________________________
Radicals and Exponents
How can we describe the pattern shown below?
•
•
•
•
•
•
64 =
16 =
81 =
225 =
64 =
729 =
82 = 8
24 = 2 2 = 4
34 = 3 2 = 9
152 = 15
26 = 2 3 = 8
36 = 33 = 27
The pattern is divide the exponent by 2 and remove the radical.
Radicals and Exponents
• Using this pattern, we can rewrite the following
•
•
•
•
𝑥8 =
𝑎4 =
𝑏12 =
𝑝2 =
• Radicals have some properties that will help us to rewrite radical
expressions in simpler forms
Radicals and Exponents
• Using this pattern, we can rewrite the following
•
𝑥8 = 𝑥
•
𝑎4
•
•
=𝑎
𝑏12 = 𝑏
𝑝2 = 𝑝
8
2
= 𝑥4
4
2
= 𝑎2
12
2
2
1
= 𝑏6
= 𝑝1 = 𝑝
• Radicals have some properties that will help us to rewrite radical
expressions in simpler forms
Radical Properties
• If 𝑎 and 𝑏 are non-negative real numbers, then 𝑎𝑏 = 𝑎 ⋅ 𝑏 and
𝑎 ⋅ 𝑏 = 𝑎𝑏
• Similarly,
𝑎
𝑏
=
𝑎
𝑏
and
𝑎
𝑏
=
𝑎
,𝑏
𝑏
≠0
• Using these properties, we can simplify the following
•
𝑥4𝑦6 = 𝑥4 ⋅ 𝑦6 = 𝑥2 ⋅ 𝑦3 = 𝑥2𝑦3
•
ℎ6
𝑔10
=
ℎ6
𝑔10
=
ℎ3
𝑔5
• We want to use these properties to be able to write radicals in exact form
Radicals in Exact Form
• A note about how the square root operator works:
• Note that 4 = 22 = 2
• But we might also consider 4 = −2 2 = −2
• This would mean that there are two possible different answers for the square
root of any number 𝑥 2 : 𝑥 2 = 𝑥 or − 𝑥
• We want the square root of a number to be a single number, so we define the
principal square root to be the positive value from above
• This means that 𝑥 2 = 𝑥 by definition (never equal to −𝑥)
Radicals in Exact Form
• If a number is a perfect square, such as 4 or 16 or 25, then its square
root is a whole number:
•
•
•
4=2
16 = 4
25 = 5
• If a number is not a perfect square, then its square root is an irrational
number
• This means that it has an infinite decimal part that cannot be written as
the ratio of two integers, so that it is impossible to write out the entire
number
Radicals in Exact Form
• Some examples of irrational square roots are
•
•
•
2
3
5
• We can write out approximate values and use these for calculations
•
•
•
2 ≈ 1.414
3 ≈ 1.732
5 ≈ 2.236
• If we want to express a square root in exact form, we leave it in the
radical or we factor it and leave part in the radical
Radicals in Exact Form
• Some numbers, though they are not perfect squares, may include one
or more perfect squares as factors
• When this occurs, we simplify the radical using the multiplication
property of radicals
• For example, since 8 is not a perfect square, then 8 cannot be fully
written out
• However, we can rewrite 8 as 23 = 22 ⋅ 2
• Now, using the multiplication property of radicals, 22 ⋅ 2 = 22 ⋅
2=2 2
Radicals in Exact Form
• Use your prime factorization chart to rewrite the following
•
•
•
•
48
56
128
135
Radicals in Exact Form
• Use your prime factorization chart to rewrite the following
•
•
•
•
48 =
56 =
128 =
135 =
24 ⋅ 3
23 ⋅ 7
27
33 ⋅ 5
• Now, we want to use the multiplication property of radicals to factor
each radical such that the left radical has only even exponents and the
right radical has only odd exponents
Radicals in Exact Form
• Use your prime factorization chart to rewrite the following
•
•
•
•
48 =
56 =
128 =
135 =
24 ⋅ 3 = 24 ⋅ 3
23 ⋅ 7 = 22 2 ⋅ 7 = 22 ⋅ 2 ⋅ 7
27 = 26 ⋅ 2 = 26 ⋅ 2
33 ⋅ 5 = (32 )(3 ⋅ 5) = 32 ⋅ 3 ⋅ 5
Radicals in Exact Form
• Note that the left radical is a perfect square, so we can simplify as
follows
•
•
•
•
48 =
56 =
128 =
135 =
24 ⋅ 3 = 24 ⋅ 3 = 22 3 = 4 3
23 ⋅ 7 = 22 ⋅ 2 ⋅ 7 = 22 ⋅ 2 ⋅ 7 = 2 14
27 = 26 ⋅ 2 = 26 ⋅ 2 = 23 2 = 8 2
33 ⋅ 5 = 32 3 ⋅ 5 = 32 ⋅ 3 ⋅ 5 = 3 15
Radicals in Exact Form
• We can also simplify when the expression in the radical (called the
radicand) includes both numbers and variables
• For example
•
20𝑥 3
•
72𝑦 7
•
144𝑚9
Radicals in Exact Form
• We can also simplify when the expression in the radical (called the
radicand) includes both numbers and variables
• For example
•
20𝑥 3 = 22 ⋅ 5 ⋅ 𝑥 3 =
22 ⋅ 𝑥 2 5𝑥 = 22 ⋅ 𝑥 2 ⋅ 5𝑥 = 2𝑥 5𝑥
•
72𝑦 7 =
•
144𝑚9 = 24 ⋅ 32 ⋅ 𝑚9 = 24 ⋅ 32 ⋅ 𝑚8 𝑚 =
24 ⋅ 32 ⋅ 𝑚8 ⋅ 𝑚 = 22 ⋅ 3 ⋅ 𝑚4 𝑚 = 12𝑚4 𝑚
23 ⋅ 32 ⋅ 𝑦 7 = 22 ⋅ 32 ⋅ 𝑦 6 2𝑦 =
22 ⋅ 32 ⋅ 𝑦 6 ⋅ 2𝑦 = 2 ⋅ 3 ⋅ 𝑦 3 2𝑦 = 6𝑦 3 2𝑦
Guided Practice
Simplify the following radicals.
a)
148𝑥 6
b) 3𝑥 30𝑥 5
c) −2𝑥 75𝑥 3
d) 6 72𝑥
e) −𝑥 2 54𝑥 5
Guided Practice
Simplify the following radicals.
a)
148𝑥 6 = 2𝑥 2 37
b) 3𝑥 30𝑥 5 = 3𝑥 2 30𝑥
c) −2𝑥 75𝑥 3 = −10𝑥 2 3𝑥
d) 6 72𝑥 = 36 2𝑥
e) −𝑥 2 54𝑥 5 = −3𝑥 4 2𝑥
Practice 7
• Handout