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Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Welcome to Math 19500 Video Lessons
Prof. Stanley Ocken
Department of Mathematics
The City College of New York
Fall 2013
An important feature of the following Beamer slide presentations is that you, the
reader, move step-by-step and at your own pace through these notes. To do so, use
the arrow keys or the mouse to move from slide to slide, forwards or backwards. Also
use the index dots at the top of this slide (or the index at the left, accessible from the
Adobe Acrobat Toolbar) to access the different sections of this document.
To prepare for the Chapter 1.2 Quiz (September 16th at the start of class), please
Read all the following material carefully, especially the included Examples.
Memorize and understand all included Definitions and Procedures.
Work out the Exercises section, which explains how to check your answers.
Do the Quiz Review and check your answers by referring back to the Examples.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Powers of numbers and letters.
Powers help us write and understand large numbers. Let’s begin with powers of 10.
102 means the product of two 10’s. It’s easy to understand 100 = 102 = 10 · 10.
It’s much harder to think about the number 100, 000, 000, 000, 000.
But if we rewrite it as 1014 , the product of 14 tens, it’s easier to handle.
You know that 100 = 102 , 1000 = 103 , and 1000000 = 106 .
Similarly, 1014 is the decimal number consisting of 1 followed by 14 zeros.
To formulate basic rules for powers, copy the behavior of powers of 10.
Product of powers rule: 10m · 10n = 10m+n .
Example: 100 · 1000 = 100000 because 102 · 103 = 102+3 = 105 .
Power to a power rule: (10m )n = 10mn .
Example: (102 )3 = (100)3 = 100 · 100 · 100 = 1000000 because (102 )3 = 102·3 = 106 .
Power of a product rule: (8 · 9)m = 8m · 9m
Example: (8 · 9)2 = (8 · 9) · (8 · 9) = (8 · 8) · (9 · 9) = 82 · 92
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
We like powers of ten because they are the basis for writing decimal numbers.
To talk about other powers, we need to define the parts of a power expression.
106 is the 6th power of 2. In the symbol 106 , 10 is the base and 6 is the exponent.
However, some people refer to 6 as the power.
Power expressions don’t need to involve numbers. For example xm is the mth power of
x. If m = 4, then x4 = xxxx, the product of 4 x’s. We say: x4 is the 4th power of x.
We rewrite the three rules from the previous slide as
Basic power identities
Product of powers identity: xm xn = xm+n .
To rewrite the product of powers of the same base, add exponents.
Power of power identity: (xm )n = xmn .
To rewrite a power of a power, multiply exponents
Power of product identity: (xy)m = xm y m .
A power of a product equals the product of powers.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Since these are identities, we can substitute any expressions for x, m and n. For the
time being, however, we assume that m and n are whole numbers.
For example: substitute 2x + 3 for x, 2x + 2 for y, 7 for m, and 5 for n:
Product of powers of same base: (2x + 3)7 (2x + 3)5 = (2x + 3)7+5 = (2x + 3)12 ;
Power of a power:((2x + 3)7 )5 = (2x + 3)7·5 = (2x + 3)35 ;
Power of a product: ((2x + 3)(2x + 2))7 = (2x + 3)7 (2x + 2)7 .
Let’s figure out an identity about dividing powers by looking at two examples:
x5
3
5−2
xxxx = xxx
= x1·
1 =x =x
x2
xx
x2
x5
x2
x5
1
xx = xxx
=
= x1·
= x13 .
xxxx
The first example shows the following:
Procedure
xm
xn
= xm−n To divide powers, subtract exponents:
Does this procedure work for the second example?
2
Substituting 2 for m and 5 for n gives xx5 = x2−5 = x−3 .
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
But we already know that
x2
x5
Radicals and fractional exponents
1
,
x3
=
Simplifying radicals.
and this suggests that x−3 =
Exercises
Quiz Review
1
.
x3
Definition of negative powers
x−n =
1
xn
This definition is an identity. To substitute an expression for x, use parentheses. Here
are some examples.
1
Substitute (x2 + 7) for x to get (x2 + 7)−1 = 2
.
x +7
1
Substitute (xy + 3) for x to get(xy + 3)−5 =
(xy + 3)5
You probably already know that x0 = 1 and x1 = x, as well as identities involving
involving fractions that will follow easily once we have introduced fraction notation. All
the basic identities are listed on the next slide for your convenience. In those identities,
m and n can now be any integers, positive or negative. We will see later that they
make sense and remain true if m and n are allowed to be fractions.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Power identities
x0 = 1
x1 = x
xm xn = xm+n
xm
= xm−n
xn
(xm )n = xmn
(xy)m = xm y m
1
x−n = n
x
m
x
xm
= n
y
y
−1
x
y
=
y
x
−m x
y m
ym
=
= m
y
x
x
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Exercise: Rewrite and verify all the above identities with m = 2; n = 3; x = 4, and
y = 2.
Some incorrect identities involving whole number powers
Make sure you don’t confuse the following two examples:
2
x3 x2 = x3+2 = x5 . However x3 = x3·2 = x6 .
Here are some additional gentle warnings:
xm + xn does not simplify (unless m = n. In that case, xm + xm = 2xm ).
(x + y)m is not xm + y m .
(x + y)m can be rewritten as a simplified sum if m = 2, 3, 4, ..., but you need to
multiply out to figure out the answer. For example
(x + y)3 = (x + y)(x + y)2 = (x + y)(x2 + 2xy + y 2 ) = x3 + 3x2 y + 3xy 2 + y 3
In particular, (x + y)3 is not x3 + y 3 .
Please don’t invent identities!
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Rewriting fractions that involve powers of variables
The basic fraction identity
1
A
=A·
= A · B −1 .
B
B
Numerical example:
7
1
= 7 · = 7 · 6−1
6
6
Every identity involving fractions is derived from the above statement. For example,
substituting x4 for A and y 5 for B gives
(x4 )
1
= (x4 ) · 5 = (x4 ) · (y 5 )−1 = (x4 ) · (y −5 ) = x4 y −5
5
(y )
(y )
The last step applied the power to a power rule and omitted useless parentheses.
24
This is a pattern that you need to know in your bones. For example, 5 = 24 · 3−5 .
3
Two methods for rewriting products of powers:
An expression that is a product of powers can be written in two ways:
as a fraction with only positive powers, or
without a fraction, but possibly including negative powers.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
a5 b5
without a fraction line.
c6
1
a5 b5
Solution: 6 = a5 b5 · 6 = a5 b5 c−6 .
c
c
1
Example 2: Rewrite 6 7 without a fraction line.
c d
1
Solution: Start out with
c6 d7
Example 1: Rewrite
Definition of negative powers:
Power of product identity
Power to a power identity
Example 3: Write
= (c6 d7 )−1
= (c6 )−1 (d7 )−1
= c−6 d−7 . This is the answer.
a5 b5
without a fraction line.
c6 d7
a5 b5
= (a5 b5 )(c6 d7 )−1 = a5 b5 c−6 d−7
c6 d7
The critical move was to apply the power of a product identity by substituting c6 for
A, d7 for B, and −1 for m = −1 in (AB)m = Am B m to see that (c6 d7 )−1 = c−6 d−7 .
Solution:
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
You need to perform this move in your head. Some more examples:
•
(a5 b6 )2 = a5·2 b6·2 = a10 b12
•
(a5 b6 c−8 )−2 = a−10 b−12 c16
Procedure
Moving powers between numerator and denominator:
If a power is a factor of a fraction’s numerator or denominator, you may rewrite that
fraction by using the following identities:
E
xm E
E
x−m E
= m
and
= −m
F
x F
F
x F
A proof of the first identity:
Example 4:
x−m E
=
F
x−5
x−m ·
E
1
F = xm
y3
x−12
y3
= 12 and −3 −4 = 5 −4 =
−3
y
x
y x
x x
·
E
F
=
E
xm F
y3
x
y5
y5
x−3 y 5
=
=
z3
x3 (z 3 )
x3 z 3
Warning: In the above examples, xm or x−m is a factor of the numerator or
denominator. If that’s not true, the power can’t be moved across the fraction line.
Here are two examples:
Example 5:
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
−2
Example 6: x y+xy
cannot be rewritten as
3
−2
numerator x + xy.
Example 7: In
a+6
,
c−2 (d8 )
xy
x2 y 3
Simplifying radicals.
Exercises
Quiz Review
because x−2 is not a factor of the
you can rewrite c−2 in the denominator as c2 in the
numerator. However, you must rewrite this fraction as
c2 (a+6)
,
d8
not as
c2 a+6
.
d8
To see why, use parentheses to substitute c for x, 2 for m, a + 6 for E, and d8 for F in
m
c2 (a+6)
E
. Parentheses are necessary!
= x F E to get c(a+6)
−2 (d8 ) =
x−m F
d8
2 2
without parentheses.
Example 8: Rewrite b2a
3 c4
2 2
2a
Solution (long version): The original problem is
b3 c4
2
(2a2 )
A m
Am
2
3 4
Use B
=B
= (b
m with A = 2a and B = b c to get
3 c4 )2
2
Use (AB)2 = A2 B 2 to rewrite the numerator
=
Use the same identity to rewrite the denominator
=
Use (Am )n = (A)mn to get the answer:
=
22 (a2 )
(b3 c4 )2
2
Stanley Ocken
22 (a2 )
(b3 )2 (c4 )2
4a4
b6 c8
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
−1 −3
b
Example 9: Rewrite 2a
as a reduced fraction without negative exponents.
a2 b−3
You can’t rush into this problem. The most important principle is that you are trying
to get rid of negative powers. This can be done is several ways.
Strategy 1: Rewrite negative power factors across the fraction line as positive powers.
−1 −3
2a b
Here is the original problem
a2 b−3
3 −3
Rewrite a−1 and b−3 across the fraction line = a2bb
2 a1
4 −3
Apply power times power rule = 2b
a3
3 3
a
Apply (A/B)−3 = (B/A)3 = 2b
4
Apply (A/B)3 = A3 /B 3
=
(a3 )3
(2b4 )3
Apply (Am )n = Amn
=
a9
23 (b4 )3
Stanley Ocken
=
a9
8b12
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
After a while, you should be able to skip the explanatory remarks and write just:
−1 −3 3 −3 4 −3 3 3
3
9
(a3 )
2a b
2b b
2b
a
a9
=
=
=
=
= 23 (b
= 8ba12
4 )3
a2 b−3
aa2
a3
2b4
(2b4 )3
Strategy 2: First flip the fraction to eliminate the outside negative power.
−1 −3 2 −3 3
2 )3 (b−3 )3
2a b
a b
a6 b−9
a6 a3
a9
= 2a
= (a
= 8a
−1 b
−3 b3 = 8b3 b9 =
a2 b−3
23 (a−1 )3 b3
8b12
Strategy 3: Combine the last two methods. First eliminate the outside negative power.
Next eliminate all inside negative powers.
−1 −3 2 −3 3 2 3 3 3
(a3 )3
a b
a a
a
a9
2a b
= 2a
= 2bb
= 2b
= (2b
−1 b
3
4
4 )3 =
a2 b−3
8b12
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Algebra and Geometry: Powers and measurement
If a square has side length s, its area is s · s = s2 . Thus A = s2 . For example, a square
with side length 5 centimeters (cm) has area 52 = 25 square centimeters (cm3 ). If we
know s, it’s easy to find A.
These statements are illustrated by the square at the right, with
side length 5 and area 52 = 25. As stated in the introduction, we
will usually not mention specific units of measurement.
Now let’s go in reverse. Suppose we know the area A and would like to figure out the
side length s. In other words, we need to solve the equation
s2 = A for the side length
√
s, which must be a positive real number. Then s = A, the square root of A.
√
For example, the side length of a square with area 9 is 9 = 3. In general, the side
length will be irrational. For example, the side length of a square
with area 2 is
√
√
2 ≈ 1.414 while the side length of a square with area 3 is 3 ≈ 1.732 .
A similar discussion applies to a cube with
side length s. Its volume is V = s3 . If we
√
3
know V and want to find s, then s = V is called the cube root of V.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Radical notation and fractional exponents
In the last slide, the powers and roots involved were positive real numbers, used for
measurement. But it’s tricky to extend the discussion to negative numbers.
• The equation x2 = 4 has two solutions x = 2 and x√= −2, written x = ±2.
• The square root of 4 is the positive solution. Thus 4 = 2, not ±2.
• The equation
x2 = −4 has no solutions, since the square of any real number is
√
positive. Thus −4 is undefined.
Squares and square roots
•
•
For all real numbers u > 0, the equation x2 = u has two solutions:
√
√
x = u (positive) and x = − u (negative).
√
For all real numbers u < 0, x2 = u has no solutions and so u is undefined.
The situation is completely different when we deal with third powers and roots.
Cubes and cube roots
For all real numbers u, the equation x3 = u has one and only one solution x =
which has the same sign as u.
Stanley Ocken
√
3
M19500 Precalculus Chapter 1.2: Exponents and Radicals
u,
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Example
√ 10:
4=2
•
√
• The solution of x2 = 4 is x = ± 4 = ±2.
• However, x2 = −4 has no solutions.
√
8 = 2.
• x3 = 8 has one solution x = 3 √
• x3 = −8 has one solution x = 3 −8 = −2.
Similar statements hold for solutions of xn = u depending on whether n is even or odd.
√
For n > 3, we call n u the nth root of u.
Even powers and roots
Suppose n is even.
• For all real numbers u > 0, the equation xn = u has two solutions:
√
√
x = n u (positive) and x = − n u (negative).
√
• For all real numbers u < 0, xn = u has no solutions and so n u is undefined.
Odd powers and roots
Assume n is odd. For all real numbers u, the equation xn = u has one and only one
√
solution x = n u, which has the same sign as u.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Example 11:
The pattern
for odd roots:
√
3
8 = 2 because 23 = 8.
•
√
3 = 8 is x = 3 8 = 2.
• The
solution
of
x
√
3
−8 = −2 because (−2)3 = 8. √
•
• The solution of x3 = −8 is x = 3 −8 = −2.
The pattern
for even roots:
√
4
81 = 3 because 34 = 81 and 3 is positive. √
•
• The
equation x4 = 81 has two solutions x = ± 4 81 = ±3.
√
4
•
−81 is undefined because x4 = −81 has no solution. That’s because
x4 = x2 · x2 , the product of two positive numbers, is positive.
Special advice involving numerical expressions with radicals:
√ √
√
Expressions such as 1, 4, and 9 should be rewritten immediately as 1, 2, and 3,
respectively. I am emphasizing this point, since you may have been told otherwise in
your high school courses.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Until now, xn has been defined only when n is an integer, positive or negative. We now
show that the roots we have been discussing can be represented as fractional powers.
In the power to a power rule (um )n = umn , it was assumed that m and n are whole
numbers. If, however, we write m = 1/n, then the power to a power rule says
(u1/n )n = u1/n·n = u1 = u. In other words:
• x = u1/n is a solution of xn = u. However, we have seen that
√
• x = n u is a solution of xn = u.
√
• Conclusion: u1/n = n u
The above conclusion is not entirely correct. After all, we have usually understood that
variables represent real numbers, although we have not stated this explicitly. If n is
√
even, then u1/n = u is defined if u < 0. However, if variables represent positive
number, than we can be comfortable writing identities with fractional powers as on the
next slide.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Identities involving radicals are obtained by rewriting power identities.
For instance, if we use m = 1/2 to rewrite (xy)m = xm y m ,
√ √
√
we get (xy)1/2 = x1/2 y 1/2 and so xy = x y.
Square root identities
√ √
√
xy = x y if x ≥ 0 and y ≥ 0.
q
√
x
x
√ if x ≥ 0 and y > 0.
=
y
y
√
√
x + y = x + y is NOT an identity.
√
√ √
√
Example 12: 12 = 4 3 = 2 3
Warning:
√
Example 13:
√3
12
=
√ 3√
4 3
=
3
√
.
2 3
We usually prefer to write a fraction without a
radical sign in the denominator. In this case, multiply by
solution as follows:
√3
12
=
3
√
2 3
=
√
3 √
3
√
2 3 3
=
Stanley Ocken
√
3 3
2(3)
√
=
√
√3
3
to obtain a complete
3
2 .
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review
Welcome
Whole number powers
Rewriting Fractions
We defined above x1/n =
√
n
Radicals and fractional exponents
m
n
Simplifying radicals.
Exercises
1
1
n = n
xm·(1/n)
· m we can substitute 1/n for n
√
in the identity
=
to obtain x =
= (xm )1/n = n xm . The
following summarizes this statement and related ideas.
(xm )n
xmn
x . Since
=m·
Quiz Review
m
n
Identities with fractional powers and radicals
Assume m and n are integers.
√
n
xn = x if n is odd or x ≥ 0 .
√
n
xn = |x| if n is even.
√
m
1
(x) n = (xm ) n = n xm if n is odd or x ≥ 0.
1 m
m
√ m
(x) n = x n
= ( n x) if n is odd or x ≥ 0.
√ √
√
√
√
n xy = n x n y if n x and n y are both defined.
q
√
n
n x = √x if both nth roots are defined and y is not 0.
ny
y
.
Exercise: Please check the last two identities by letting n = 2 and x = 4 or −4.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Example 14:
√
1
1
3
2 4 = 2 4 ·3 = (2 4 )3 = ( 4 2)3 . Another possibility:
√
1
3
1
2 4 = 23· 4 = 23 4 = 4 8. This answer looks nicer.
1 3
√
√ 3
3
4
x 4 = x 4 = ( 4 x) or x3
1 2
2
27 3 = 27 3 = 32 = 9
3
16− 4 =
1
3
16 4
=
1
1 3
16 4
=
1
23
=
1
8
Definition
√
√
√
√
In radical expressions such as 12 or a + b or 5 12 or 5 a + b, the radicand is the
number or expression (12 or a + b in these examples) under the radical sign.
√
√
√
Warning: n x + y IS NOT equal to n x + n y.
√
√
√
In particular x + y IS NOT equal to x + y.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Simplifying expressions with radicals.
√ √
√
The rule xy = x y (for x and y positive) allows us to simplify square root
expressions.
√
√
√ √
√
• √12 = 4√· 3 = 4 3√= 2 √3
√
Example 15:
1000 = 100 · 10 = 100 10 = 10 10
•
Procedure
To simplify a square root, factor out from the radicand the largest possible number
that is a perfect square. You can do this by trial and error or by using the prime power
factorization of the radicand.
In all cases the strategy is to factor our squares from the radicand.
Example 16: Simplify each of the following.
√
√
9 = 32 = 3
√
√
√ √
√
√
18 = 2 · 32 = 2 32 = 2 · 3 = 3 2
√
√
√ √
√
45 = 9 · 5 = 9 5 = 3 5
√
√
√ √ √
√
√
108 = 4 · 27 = 4 9 · 3 = 2 · 3 · 3 = 6 3
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
The general strategy is to factor out the highest possible even power from each power
in the radicand.
√
√
Example 17: 211 313 52 = 210 · 2 · 312 · 3 · 52
√ √ √ √ √
√
√
√
= 210 2 312 3 52 = 25 · 2 · 36 · 3 · 5 = 25 · 36 · 5 6
√
√
√
√ √ √
√
√
Example 18: 288 = 32 · 9 = 16 · 2 · 9 = 16 2 9 = 4 · 2 · 3 = 12 2
√
√
Example 19: Simplify 3 216 + 5 96
Solution:
out the square
√
√First work √
√roots:
√ √ √ √
√
√
216 = 2 · 108 = 2 · 4 · 27 = 2 · 4 · 3 · 9 = 2 4 3 9 = √
2·√
2 · 3 ·√
3
=
6
2
3
=
6
6
√
√
√ √
√
Or, a bit faster: 216 = 36 · 6 = 36 6 = 6 6
√
√
√ √
√
Similarly, 96 = 16 · 6 = 16 6 = 4 6
Then complete the original problem:
√
√
√
√
√
√
√
3 216 + 5 96 = 3 · 6 6 + 5 · 4 6 = 18 6 + 20 6 = 38 6
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
Here
√ are examples with variables. The strategy is still to factor out even powers, since
x2n√= (x2n )1/2 = (x)2n·1/2 = xn . To avoid technical difficulties arising from the fact
that x2 = |x|, not x, assume all variables represent positive numbers.
√
√
√ √
√
Example 20: x21 = x20 · x = x20 x = x10 x
p
p
√ p √
√
Example 21: x11 y 13 = x10 y 12 xy = x10 y 12 xy = x5 y 6 xy
p
p
Example 22: Simplify x3 y 3 x11 y 11 + 7x2 y x13 y 15 .
Again we simplify the radicals separately:
p
p
p
√ p √
√
√
√
x11 y 11 = x10 y 10 xy = x5 y 5 xy and x13 y 15 = x12 y 14 xy = x6 y 7 xy .
Therefore
p
p
x3 y 3 x11 y 11 + 7x2 y x13 y 15
√
√
= x3 y 3 x5 y 5 xy + 7x2 yx6 y 7 xy
√
√
√
= 1x8 y 8 xy + 7x8 y 8 xy = 8x8 y 8 xy
A similar strategy works for cube roots: a power xn has a nice cube root if n is a
multiple of 3.
√
1/3
3
Example 23: x30 = (x30 ) = (x)30·1/3 = x10
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
A power of a variable is a perfect cube provided that power is a multiple of 3.
In the following we factor out the largest power of x that is a multiple of 3
Example 24:
√
√
√
√
√
3
3
3
x31 = x30 x = x30 3 x = x10 3 x
√
√
√
√
√
3
3
3
3
3
x32 = x30 x2 = x30 x2 = x10 x2
√
p
3
x33 = 3 (x11 )3 = x11
Finally, we do some problems involving rewriting powers with fractional exponents.
This doesn’t involve anything new. Again, assume variables represent positive numbers.
1/7
Example 25: Simplify (s2 ) s1/2
1/7
Solution: First find (s2 ) = (s)2·(1/7) = s2/7 . Then
1/7
(s2 ) s1/2 = s2/7 s1/2 = (s)2/7+1/2 = s11/14 since
2
1
2 2
1 7
4
7
11
7 + 2 = 7 · 2 + 2 · 7 = 14 + 14 = 14 .
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Welcome
Whole number powers
Rewriting Fractions
Example 26: Simplify
Radicals and fractional exponents
x8 y −4
16y 4/3
Simplifying radicals.
−1/4
Solution
8 −42:−1/4
Solution
8 −41:−1/4
x y
16y 4/3
x y
16y 4/3
−1/4
1/4
=
(x8 y−4 )
−1/4
(16y4/3 )
=
16y 4/3
x8 y −4
=
x−2 y 1
16−1/4 y −1/3
=
16y 4/3−(−4)
x8
=
16y 16/3
x8
=
=
Exercises
161/4 y 1 y 1/3
x2
2y 4/3
x2
Stanley Ocken
1/4
1/4
=
(16y 16/3 )1/4
(x8 )1/4
=
161/4 (y 16/3 )1/4
(x2
=
2y 4/3
x2
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Exercises for M19500 Chapter 1.2: Exponents and radicals
Click on
Wolfram Calculator
to find an answer checker.
Click on
Wolfram Algebra Examples
to see how to check various types of algebra problems.
1. Do the WebAssign homework.
2. Solve each of the following equations
a) x2 = 7
b) x2 = 9
c) x3 = 8
d) x4 = 16
e) x2 = −16
f) x3 = −64
3. Rewrite each of the following with neither radicals nor fraction exponents:
√
√
√
a) 9
b) −7
c) 3 27
√
√
1
d) 3 −27 e) 4 16
f) 9− 2
3
g) 27−2/3 h) (−27)−2/3 i) (−27)− 2
√
√
√
√
√
√
b) 500 + 125 + 605
4. Simplify a) 32 + 50 + 200
5. Rewrite without negative exponents:
1/6 −3 3 −2 −1 −2 −1 −2 1/6 −3 −3
x b
x b
a b
a) a x−1by
b)
3/2
1/3
x−1 y
a y
a3/2 y 1/3
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
6. Rewrite without negative exponents or radical signs:
Simplifying radicals.
Exercises
p
p
5
x3 y 2 10 x4 y 16
7. Rewrite√as a single term with
√ lowest possible powers in the radicand:
a) 4a3 b2 c 12a5 b7 c9 − 3a4 b4 c4 27a3 b3 c3
√
√
b) 4a3 x2 c 12a9 x1 1c9 − 3a4 x4 c4 27a7 x7 c3
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Quiz Review
a5 b5
without a fraction line.
c6
1
Example 2: Rewrite 6 7 without a fraction line.
c d
a5 b5
Example 3: Write 6 7 without a fraction line.
c d
x−5
x−12
Example 4: Rewrite −3 and −3 −4 without negative exponents.
y
y x
Example 1: Rewrite
Example 5: Rewrite
Example 6: Does
x−3 y 5
without negative exponents.
z3
x−2 +xy
y3
Example 7: Rewrite
Example 8: Rewrite
a+6
c−2 (d8 )
2a2
b 3 c4
equal
xy
x2 y 3
? Why or why not?
without negative exponents.
2
without parentheses.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review
Welcome
Whole number powers
Example 9: Rewrite
Rewriting Fractions
2a−1 b
a2 b−3
−3
Radicals and fractional exponents
Simplifying radicals.
Exercises
as a reduced fraction without negative exponents.
√
Example 10:
•
4=?
• The solution of x2 = 4 is x =?
• Solve x2 = −4.
• Solve x3 = −8.
Example 11:
√
• Find 3 8.
• Solve √
x3 = 8.
3
• Find −8.
• Solve √
x3 = −8.
• Find 4 81.
• Solve √
x4 = 81.
4
• Find −81.
Example 12: Rewrite
Example 13: Rewrite
√
12 by factoring squares out of the radicand.
√3
12
without a radical in the denominator.
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
Example 14: Rewrite each expression without fractional powers.
3
24
3
x4
2
27 3
3
16− 4
Example
√ 15: Simplify the following square roots:
• √12.
•
1000.
Example 16: Simplify each of the following.
√
9.
√
18.
√
45.
√
108
√
Example 17: Simplify 211 313 52 .
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review
Welcome
Whole number powers
Rewriting Fractions
Radicals and fractional exponents
Simplifying radicals.
Exercises
√
288.
√
√
Simplify 3 216 + 5 96.
√
Simplify x21 .
p
Simplify x11 y 13 .
p
p
Simplify x3 y 3 x11 y 11 + 7x2 y x13 y 15 .
√
3
Simplify x30 .
Example 18: Simplify
Example 19:
Example 20:
Example 21:
Example 22:
Example 23:
Example 24: Simplify each of the following:
√
3
x31 .
√
3
x32 .
√
3
x33 .
1/7
Example 25: Simplify (s2 ) s1/2 .
8 −4 −1/4
x y
Example 26: Simplify
.
16y 4/3
Stanley Ocken
M19500 Precalculus Chapter 1.2: Exponents and Radicals
Quiz Review