Download Section 10.1: Radical Expressions and Functions

Math 23 - Section 10.1 - Radical Expressions and Functions. - Page 1
Section 10.1
Radical Expressions and Functions
I.
Remember
A To square a number, we multiply the number by itself, the result is called the square
of the original number.
B. Examples
1.
(11)2 = (11)(11) = 121
2.
(−13)2 = (−13)( −13) = 169
II.
Square Root
A. To find the square root of a number, we undo the process of squaring, i.e. – We will
start with a number and to find its square root, we find what number times itself
gives us the original number.
B. Definitions
1.
In the expression
25 , “25” is called the radicand, "
" is called the
radical sign, and the whole thing is called a radical expression.
2.
If x is any positive real number, then x is called the positive square root
of x. This means that it will always be POSITIVE.
− x is called the negative square root of x and will always be NEGATIVE.
Note that any positive real number will have two square roots, a positive and
a negative one.
5.
If a number is a perfect square, its square root is a rational number. If a
number is not a perfect square, its square root is an irrational number.
6.
At this time, we will not be trying to find the square root of any negative
numbers. So for now, we will say that the square root of a negative number is
not a real number.
Examples – Find the following square roots.
3.
4.
C.
1.
− 1681
Since we have a negative sign in front of the radical sign, we are looking
for the negative square root of 1681.
Using the square root key on our calculators, we get:
Answer: −41
2.
− 169
Since this is the square root of a negative number, our answer is not a
real number.
Answer: Not a real number.
3.
Now you try one:
256
Answer: 16
144 + 25
The radical sign is a grouping symbol, so we have to obey the Order of
Operations, which means we have to simplify inside the radical first, by adding
the numbers.
144 + 25 = 169
Now find the square root.
4.
Answer:
13
© Copyright 2005by John Fetcho. All rights reserved
Math 23 - Section 10.1 - Radical Expressions and Functions. - Page 2
5.
144 + 25
On this one, we need to take the square root first, then add.
144 + 25 = 12 + 5
Answer:
17
Note that the order matters!
6.
Now you try one:
Answer:
7.
25 − 16
3
( x − 2)
2
On this one, we have to recognize that when it comes to exponents, even
exponents are perfect squares. To take the square root of an exponential, we
keep the base but divide the exponent by 2. Also, this notation means that
we want the positive square root, so that means that we have to ensure that
our answer is positive.
Answer:
8.
− 49x 6
On this problem, we want our answer to be negative since we want the
negative square root. Then, we will take the square root of the coefficient.
Next, on the exponent, the square root will be 6 divided by 2. Finally, we
have to ensure that our overall answer is negative, so we have to remember
that if x is a negative number, x3 will also be negative; then we would have a
negative times a negative, which is a positive. So we need to use absolute
value bars on the x3 to make our final answer a negative times a positive.
Answer:
9.
−7|x3|
Now you try one:
Answer:
III.
|x−2|
x 2 + 12 x + 36
|x + 6|
Cube Roots and Higher
A. Just as we can cube a number (multiply itself three times), we can undo this to find
the cube root of a number. For example, since 23 = 8, the cube root of 8 is 2, which
we will write as:
3
8=2
The “3” is called the index and tells us what root we are looking for.
So 4 81 is read “the fourth root of 81” and means that we are looking for the number
which multiplied by itself four times will give us 81 (it’s 3).
© Copyright 2005by John Fetcho. All rights reserved
Math 23 - Section 10.1 - Radical Expressions and Functions. - Page 3
B.
Examples – Simplify each radical expression.
3
1.
− 125
Since we have an odd index, the answer is a real number.
We need to determine what negative number, times itself 3 times, is equal to
–125.
Answer: -5
− 3 − 125
2.
Since we just found the cube root of –125 (−5), we now want the negative of
that:
−(−5)
Answer: 5
3.
Now you try one:
− 3 27
Answer: −3
4.
3
1
1000
This looks difficult because we have a fraction! However, we are going
to break this problem up into two parts:
What is the cube root of the numerator?
What is the cube root of the denominator?
3
1 =1
3
1000 = 10
So putting this all together as a single fraction.
Answer:
5.
1
10
Now you try one:
Answer:
3
1
125
1
5
© Copyright 2005by John Fetcho. All rights reserved