Transcript Cube Root - Web4students

Copyright © 2012 Pearson Education, Inc.
Slide 7- 1
7.1
Radical
Expressions,
Functions, and
Models
■
Square Roots and Square-Root
Functions
■
■
■
■
Expressions of the form a2
Cube roots
Odd and Even nth Roots
Radical Functions and Models
Copyright © 2012 Pearson Education, Inc.
Square Roots and SquareRoot Functions
When a number is multiplied by itself,
we say that the number is squared.
Often we need to know what number
was squared in order to produce some
value a. If such a number can be found,
we call that number a square root of a.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 3
Square Root
The number c is a square root of a if c 2 = a.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 4
For example,
16 has –4 and 4 as square roots because
(–4)2 = 16 and 42 = 16.
–9 does not have a real-number square
root because there is no real number c
for which c 2 = –9.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 5
Example
Find the two square roots of 49.
Solution
The square roots are 7 and –7, because 72 =49
and (–7)2 = 49.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 6
Whenever we refer to the square root of
a number, we mean the nonnegative
square root of that number. This is often
referred to as the principal square root
of the number.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 7
Principal Square Root
The principal square root of a nonnegative
number is its nonnegative square root. The
symbol
is called a radical sign and is used
to indicate the principal square root of the
number over which it appears.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 8
Example
Simplify each of the following.
a)
81
16
b) 
81
Solution
a)
81  9
16
4
b) 

81
9
Copyright © 2012 Pearson Education, Inc.
Slide 7- 9
a is also read as “the square root of a,”
“root a,” or “radical a.” Any expression
in which a radical sign appears is called a
radical expression. The following are
examples of radical expressions:
12,
3m  2, and
3x  2 x
 3.
7
2
The expression under the radical sign is
called the radicand.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 10
Expressions of the Form a
2
It is tempting to write a 2  a, but the next
example shows that, as a rule, this is untrue.
Example
2
a)
8  64  8
b)
(8)2  64  8
Copyright © 2012 Pearson Education, Inc.
( ( 8) 2  8)
Slide 7- 11
Simplifying
a
2
For any real number a,
a  a.
2
(The principal square root of a2 is the absolute
value of a.)
Copyright © 2012 Pearson Education, Inc.
Slide 7- 12
Example
Simplify each expression. Assume that the
variable can represent any real number.
a)
( y  3)
b)
12
c)
2
m
10
x
Solution
a)
2
( y  3)  y  3
Since y + 3 might be negative,
absolute-value notation is
necessary.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 13
Solution continued
b) Note that (m6)2 = m12 and that m6 is
never negative. Thus,
12
m
6
m .
c) Note that (x5)2 = x10 and that x5 might
be negative. Thus,
10
x
5
 x .
Copyright © 2012 Pearson Education, Inc.
Slide 7- 14
Cube Roots
We often need to know what number was
cubed in order to produce a certain value.
When such a number is found, we say that
we have found the cube root.
For example, 3 is the cube root of 27
because 33 =27.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 15
Cube Root
The number c is a cube root of a if c 3 = a. In
symbols, we write 3 a to denote the cube root of
a.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 16
Example
Simplify
3
27 x3 .
Solution
3
27 x3  3x
Since (–3x)(–3x)(–3x) = –27x3
Copyright © 2012 Pearson Education, Inc.
Slide 7- 17
Odd and Even nth Roots
The fourth root of a number a is the number
c for which c4 = a. We write n a for the nth
root. The number n is called the index
(plural, indices). When the index is 2, we
do not write it.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 18
Example Find each of the following.
a) 5 243
b) 5 243
c)
11 11
m
Solution
5
a) 243  3
Since 35 = 243
5
b) 243  3
Since (–3)5 = –243
c)
11 11
m
m
Copyright © 2012 Pearson Education, Inc.
Slide 7- 19
Example Find each of the following.
a) 4 81
b) 4 81
c)
4
16m
4
Solution
a) 4 81  3
Since 34 = 81
b) 4 81  can't be simplified.
c)
4
16m4  2m or 2 m
Not a real number
Use absolute-value notation
since m could represent a
negative number
Copyright © 2012 Pearson Education, Inc.
Slide 7- 20
Simplifying nth Roots
n
a
Positive
n
a
n
a
n
Positive
Even
a
Negative
Positive
Not a real
number
Positive
Odd
Negative
Negative
Copyright © 2012 Pearson Education, Inc.
a
Slide 7- 21
Radical Functions and Models
A radical function is a function that can be described
by a radical expression.
If a function is given by a radical expression with an
odd index, the domain is the set of all real numbers. If
a function is given by a radical expression with an
even index, the domain is the set of replacements for
which the radicand is nonnegative.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 22
Example
Find the domain of the function given by the equation
Check by graphing the function. Then, from the graph,
estimate the range of the function.
f ( x)  x  4
2
Solution
2
The radicand of f ( x)  x  4 is x + 4.
We must have x2  4  0
2
x 2  4.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 23
continued
The domain is (–∞,∞).
The range appears to be [2,∞).
Range
Domain
Copyright © 2012 Pearson Education, Inc.
Slide 7- 24