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Summer 2016 - Session 2
Math 1300
FUNDAMENTALS OF MATH
Section #16535
Monday - Friday, 10am-12pm
Instructor: Dr. Angelynn Alvarez
[email protected]
06/07/2016
Section 1.4 – Exponents and Radicals
Section 1.4 – Exponents and Radicals
-Let x be any number. Then a number n is an exponent if it is a
superscript of x; we write 𝑥 𝑛 .
-If n is an exponent of x, we say…
“x to the n-th power”
or “x to the n-th” for short.
-We call 𝑥 𝑛 an exponential expression, and
 the number 𝑥 is called the base.
 the number 𝑛 is called the exponent.
-When n is an integer, we define the exponential expression 𝑥 𝑛 as
follows:
𝑥𝑛 = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ … ∙ 𝑥
In other words, 𝑥 𝑛 means we just multiply the number x by itself n
times. (We will deal with fractions as exponents later.)
Examples:
42
(−𝑥 )5
𝑥6
23
A fact about even exponents and negative bases:
If the exponent n is even:
(−𝑥 )𝑛 = 𝑥 𝑛
CAUTION: One must be careful when dealing with negative signs " −
" and parentheses with even exponents!
 If there are no parentheses and your exponent is even, then
your answer will be negative.
A fact about odd exponents and negative bases:
If the exponent n is odd:
(−𝑥 )𝑛 = −𝑥 𝑛
“When evaluating an expression with a negative base and odd
exponent, the answer is always negative.”
Example: Evaluate the following expressions.
(−2)3
(−3)2
−32
−24
(−2)2
Rules for Exponential Expressions with the same base:
If 𝑥 𝑛 and 𝑥 𝑚 are two exponential expressions with the same base,
then:
(1)
𝒙𝒏 ∙ 𝒙𝒎 = 𝒙𝒎+𝒏
“When multiplying two exponential expressions, keep the same
base and add their exponents.”
(2)
𝒙𝒎
𝒙𝒏
= 𝒙𝒎−𝒏
“When dividing two exponential expressions, keep the same base
and subtract their exponents.”
(3)
(𝒙𝒏 )𝒎 = 𝒙𝒏𝒎
“If the exponential expression is raised to another exponent, just
multiply the two exponents.”
Example: Write the following as a single base and exponent. (Do
not evaluate.)
229 ∙ 211
817 ∙ 830
𝑥 90 ∙ 𝑥 7
416
48
725
715
𝑥 40
𝑥 14
(143 )8
(174 )5
(𝑥 7 )9
Rule for exponential expressions with negative exponents:
Given an exponential expression with a negative exponent, 𝑥 −𝑛 , we
can always rewrite it to have a positive exponent:
𝒙−𝒏 =
𝟏
𝒙𝒏
The rule still applies for multiple expressions inside the parentheses:
(stuff)−𝒏 =
𝟏
(stuff)𝒏
Example: Simplify. Your answer should contain no negative
exponents.
𝑥 −5
7−𝑛
(7𝑥𝑦 4 )−1
(5𝑥 3 𝑦 42 )−1
Rule for exponential expressions with different bases but
same exponent:
When raising a product of two numbers, say x and y, to an exponent,
we have:
(𝒙 ∙ 𝒚)𝒏 = 𝒙𝒏 ∙ 𝒚𝒏
Examples: Rewrite the following exponential expressions.
(4 ∙ 2)3
(2 ∙ (−3))
4
(𝑥 2 ∙ 𝑦 3 )3
(−1)2 ∙ (−3)2
Rules for exponential expressions with fractions as the base:
𝑥 𝑛
Given an exponential expression with a fraction as the base, ( ) , we
𝑦
have
(1)
𝒙 𝒏
𝒙𝒏
𝒚
𝒚𝒏
( ) =
“Raising a fraction to a power is the same as raising the
numerator to the power, and then dividing by the denominator
raised to that power.”
(2)
𝒙 𝒏
𝒚 −𝒏
𝒚
𝒙
( ) =( )
“Flipping the fraction inside the parentheses changes the sign of
the exponent.”
Example: Simplify. Your answer should contain no negative
exponents.
𝑤 5
( )
𝑥
x4
( 7)
y
−1
𝑤 −1
(
)
(4𝑥 5 )
Rule for raising expressions to the 0th power:
Given any nonzero base, x,
𝒙𝟎 = 𝟏
“Anything nonzero that is raised to the 0th power is always 1.”
Summary of Rules for Exponential Expressions:
(1)
𝑥 𝑛 ∙ 𝑥 𝑚 = 𝑥 𝑚+𝑛
(2)
𝑥𝑚
(3)
(𝑥 𝑛 )𝑚 = 𝑥 𝑛𝑚
(4)
𝑥 −𝑛 =
(5)
𝑥 𝑛 ∙ 𝑦 𝑛 = (𝑥 ∙ 𝑦 )𝑛
(6)
( ) =
(7)
( ) =( )
(8)
𝑥0 = 1
𝑥𝑛
= 𝑥 𝑚−𝑛
1
𝑥𝑛
, 𝑥𝑛 =
𝑥 𝑛
𝑥𝑛
𝑦
𝑦𝑛
𝑥 𝑛
𝑦 −𝑛
𝑦
𝑥
1
𝑥 −𝑛
We will need to use these rules to simplify expressions involving
variables, in which we need to use more than 1 rule to simplify.
Example: Simplify the following expressions. Your answers should
contain no negative exponents.
(5𝑤 −3 𝑥 )−3
(9𝑥 4 𝑦 −2 )−2
−2
𝑥
(
)
(4𝑧 −3 )
−3
10𝑥 −7
(
)
𝑤
𝑥2𝑦7𝑦3
𝑥8
𝑢12 𝑢−15 𝑤 10
𝑤 −7
𝑥 11 𝑥 −17 𝑦 15
𝑦 −8
Radicals
Recall: When n is an integer, we define the exponential expression
𝑥 ! as follows:
𝑥! = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ … ∙ 𝑥
In other words, 𝑥 ! means we just multiply the number x by itself n
times.
Question: What if 𝑛 is a fraction???
!
!
When n is a fraction, say !, we define the exponential expression 𝑥 !
as follows:
In this case, we either say..
“x to the
• The symbol
!
!
!
𝑡ℎ power” or “the n-th root of x”
is called the radical.
• The radicand is the expression inside of the radical.
The Square Root
A special (and most common) case of the n-th root is when n=2.
When 𝑛 = 2, we call the expression the square root of a number x
and write
A square root of a number, say x, is any number 𝑚 such that
𝒎 ∙ 𝒎 = 𝒎𝟐 = 𝒙
--that is: A number, say m, is a square root of 𝑥 if 𝑚 multiplied by
itself is equal to 𝑥.
Examples:
• 4 is a square root of 16 because 4(4)=16
• 10 is a square root of 100 because 10(10)=100
• 7 is NOT a square root of 81 because 7(7) ≠ 81.
Important Remark:
In general, a number 𝑥 has 2 square roots: 𝑚 and – 𝑚.
In this class (on the online quizzes, exams, etc.), we will only focus
on the positive root, 𝑚.
Important Square Roots: MEMORIZE THESE!
0=0
1=1
4=2
9=3
16 = 4
25 = 5
36 = 6
49 = 7
64 = 8
81 = 9
100 = 10
121 = 11
144 = 12
The numbers, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 are
called perfect squares. (Because their square roots are nice integers.)
Example: Simplify the following.
!
6!
36
!
25!
− 144
!
−121!
Multiplying and Dividing Square Roots:
For two numbers x and y, where 𝑥 ≥ 0, 𝑦 ≥ 0:
(1)
𝒙∙𝒚=
𝒙 ∙
𝒚
“The square root of a product is equal to the product of the square
roots.”
(2)
𝐱
𝐲
=
𝐱
𝐲
“The square root of a fraction is equal to the square root of the
numerator, divided by the square root of the denominator.”
Example: Simplify the following.
(64)
(81)
(100)
(16)
−
(49)
(36)
−
(121)
(144)
It’s quite easy to simplify square roots if the number inside the
radical is a perfect square.
Question: What if the number inside the radical is NOT a perfect
square?
Simplifying the square root of a number that is NOT a perfect
square
(1)
(2)
Write the number as a product of a perfect square and
another number (the second number does not need to be
perfect square.)
Use the rule
𝒙∙𝒚=
(3)
𝒙 ∙
𝒚
and evaluate the square roots of the perfect squares.
Repeat steps (1) and (2) several times if necessary.
*You are done simplifying once all the numbers under a square
root do not have a factor that is a perfect square.
Example: Simplify the following.
32
54
360