Download Fundamentals of Mathematics - 1.4 Exponents and Radicals

Fundamentals of Mathematics
1.4 Exponents and Radicals (Continue)
Ricky Ng
Lecture 7
September 11, 2013
Ricky Ng
Fundamentals of Mathematics
Announcements
Registration for Test 1.
Fingerprint in CASA.
Online Quiz 1 due 9/11 Wednesday.
Online Quiz 2 & 3 due 9/13 Friday.
First homework is due on 9/12.
Second homework is due on 9/14.
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Fundamentals of Mathematics
Last Time...
xm ⇥ xn = xm+n
(xm )n = xmn
x0 = 1
xn ⇥ y n = (xy)n
x
n
=
1
xn
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Fundamentals of Mathematics
Recap
Recall that
xn
xm
can be view as
xn ⇥
1
= xn ⇥ x
xm
m
xm
= xm
xn
n
= xn
m
.
So,
Rule
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Fundamentals of Mathematics
Fractions with Exponents
There are two rules to know when dealing with fractions with
exponents
Rule
Given non-zero real numbers x and y, and integer n
✓ ◆n
x
xn
= n;
y
y
and
✓ ◆
x
y
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1
=
y
.
x
1.4 Exponents and Radicals
Why??
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1.4 Exponents and Radicals
Examples
Evaluate the following:
3
5
2
1 3
2
⇥
3
4
1
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1.4 Exponents and Radicals
Popper 2: Question 1
Simplify the following:
✓ ◆
3
2
2
a) 94
b) 49
c) 34
d) 29
e) None of the above
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1.4 Exponents and Radicals
Caution
When x > 0, then
n
( x) =
(
xn ,
xn ,
n is even;
n is odd.
Why? Think about ( 1)n ...
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1.4 Exponents and Radicals
More Examples
Simplify the following with no negative exponents:
(2x3 y
2 z 5 )2
a0 +b0
(a+b)0
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1.4 Exponents and Radicals
2x6 y
6x2 y 3
⇣
a2 b 5
a 1 b2
2
⌘
3
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1.4 Exponents and Radicals
Popper 2: Question 2
Simplify the following with no negative exponent:
(x2 y 1 )2 ⇥ x 4 y
a) y 3
b) y1
c) 1
d) x3
e)
y2
x4
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1.4 Exponents and Radicals
Square Roots
Definition (Square root)
A real number y is called a square root of x if y 2 = x. In
general if x > 0, then x has two sqaure roots, one is negative
and one is positive.
The principal square root of x is the positive square root,
p
and we denote it by x.
Remark
p
If x > 0, we write
x for its negative square root, and
p
± x to denote both positive and negative square roots.
p
Note that 0 = 0, so 0 only has one square root.
p
If x < 0, then x is not a real number. In this class, we
only deal real numbers, so we do not allow that.
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1.4 Exponents and Radicals
We also write
x1/2 =
This is because
p
x.
p
(x1/2 )2 = ( x)2 = x,
which agrees with the exponent rules.
Remark
In other words, square root can be seen as a fraction
p
exponent. In fact, this is more useful than · in calculations,
as we shall see later today.
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1.4 Exponents and Radicals
Examples
Find
p
64.
Find square roots of 81.
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1.4 Exponents and Radicals
Since
p
· behaves like exponents, the exponent rules apply:
Rule
For x, y > 0,
p p
p
x y = x1/2 y 1/2 = (xy)1/2 = xy,
and
s✓ ◆ ✓ ◆
p
x
x 1/2 x1/2
x
=
= 1/2 = p .
y
y
y
y
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1.4 Exponents and Radicals
Examples
Find
p
Find
q
3⇥
p
12.
81
16 .
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1.4 Exponents and Radicals
Popper 2: Question 3
Evaluate the square roots of 81. (Note that I did not write
p
81...)
a)
b)
c)
d)
9
-9
±9
Let me grab my calculator...
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1.4 Exponents and Radicals
Popper 2: Question 4
Evaluate
a)
b)
c)
d)
e)
p
2 ⇥ (8)1/2
4
±4
8
±8
8
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1.4 Exponents and Radicals
Radicals
Here are some terminology...
Definition (Radical)
For x > 0 and n a natural number, we write
p
x1/n = n x
to denote its nth -root of x. We call x a radicand, the symbol
p
· the radical, and n the degree.
Definition (Perfect square)
p
If both x and x are natural numbers, then we call x a perfect
square. For example, 4 = 22 , 49 = 72 are perfect square; but 51
is not.
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1.4 Exponents and Radicals
Simplest Radical Form
Sometimes
p
x can be further simplified. More precisely,
Definition
p
We say x is in the simplest radical form if no factor of x
is a perfect square.
Example
p
6 is in simplest radical form.
p
But 32 is not. Why? Because 32 = 2 ⇥ 16, and 16 = 42 is
a perfect square.
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1.4 Exponents and Radicals
Question
p
If x is not in the simplest radical form, how do we simplify
further?
Rule
In this case, we can write
x=
a2
|{z}
perfect square
⇥
b
|{z}
.
some other #
Then, using the product rule
p
p
p
p
p
x = a2 ⇥ b = a2 · b = a b
p
If b is again not in the simplest radical form, repeat.
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1.4 Exponents and Radicals
Examples
Write each of the following in the simplest radical form, or as a
rational number.
p
24
541/2
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1.4 Exponents and Radicals
q
80
9
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1.4 Exponents and Radicals
Popper 2: Question 5
Write
p
48
in the simplest radical form.
a)
b)
c)
d)
e)
p
4 3
p
3 8
6
p
3 6
p
4 2
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1.4 Exponents and Radicals