Download Chapter 15 Radical Expressions and Equations Notes

Chapter 15
Radical Expressions and Equations
Notes
15.1 Introduction to Radical Expressions
The symbol
is called the square root and is defined as follows:
c2 = a
a = c only if
Sample Problem:
Solution:
Simplify
16 = 4
16
since
4 2 = 16 .
Note that every positive number has two square roots, a positive and a negative root.
For example, the square roots of 16 are 4 and -4, since 4 2 = 16 and (−4) 2 = 16 . The
symbol implies the positive root, or the principal square root. To get the negative
root, a negative sign must be used in front of the square root sign as in −
Sample Problem:
Simplify
Solution:
− 16 = −4
− 16
Student Practice: Simplify each of the following radical expressions.
1.
25
2. − 81
3.
1
4. − 36
5.
100
6. − 49
7.
− 9 (not real #.)
8.
y2
.
9.
x6
11.
x 16
13.
9 x 12
10.
w10
12.
16a 2 b 8
Tip: When taking roots of exponential expressions, keep the base and take half of the exponent.
15.2 Simplifying Radical Expressions
● SIMPLIFYING RADICAL EXPRESSIONS
A radical expression is simplified when there are no perfect square factors inside the
radical; i.e., when you take as much as you can out of the radical. Simplifying can be
done using the following rule:
PRODUCT RULE
a•b = a • b
a • b = a•b
OR
The product rule can be used to simplify radicands that are not perfect squares.
Simply factor the radicand using a perfect square as a factor.
Sample Problem 1:
Simplify
36
36 = 4 • 9 = 4 • 9 = 2 • 3 = 6
Solution:
Sample Problem 2:
Simplify
45
Solution: Since 45 is not a perfect square, we first factor 45 using a perfect square
factor. It may help to list the first few perfect square factors, which are 1,4,9,16: In our
45 = 9 • 5 = 9 • 5 = 3 • 5 = 3 5
case, we’ll use 9.
______________________________________________________________________
FOR LARGER NUMBERS. For larger numbers, it can be helpful to factor the
radicand into prime factors. To do so, divide by the smallest prime number as man times
as possible, then the next smallest, then the next until you are left with a prime number.
For each pair of identical factors, .
5
Sample Problem 2: Simplify
240
3 15
2
2
2
2
30
60
120
240
Solution:
240 = 2 • 2 • 2 • 2 • 3 • 5 = 4 • 4 • 3 • 5 = 4 • 4 • 3 • 5 = 2 • 2 • 15 = 4 15
2
2
240 = 2 • 2 • 15 = 4 15
Tip: To check your answer, multiply the outer number twice and then multiply by the
inner number to get what you started with. 4 • 4 • 5 = 240
Sample Problem 3:
Solution:
Simplify
x 13
x 13 = x 12 • x = x 12 • x = x 6 x
Student Practice: Simplify each radical expression.
1.
18
2.
50
3.
27
4.
24
5.
32
6.
72
8.
64
28.
21
9. w 8
10.
x9
11.
a 15
12.
a3
13.
a 5 b10
14.
20 x 7
15.3 Addition and Subtraction of Radical Expressions
To add/subtract two radical terms, they MUST BE LIKE TERMS. In other words,
the radicands MUST be the same. If the radicands are not the same, simplify the
radicands to make them match and then combine the coefficients and leave the
radicand alone.
4 72 + 7 8
Sample Problem:
Add
Solution:
4 72 + 7 8 = 4 36 • 2 + 7 4 • 2
= 4•6 2 + 7•2 2
= 24 2 + 14 2
= 38 2
Student Practice: Add or subtract as indicated.
1. 5 2 + 8 2
2. 2 10 x − 8 10 x
3. − 4 5 + 5 5
4.
54 + 2 24
5.
2 10 − 7 40
7. 5 2 x 2 − 2 200
6. 5 18 − 2 50 + 6 2
15.4 Multiplying and Dividing Radical Expressions
When multiplying two radical expressions, recall the product rule from before which
states that the product of two radical expressions is the radical of the product.
PRODUCT RULE
a • b = a•b
Sample Problem:
12 x 3 y • 6 x 5 y 4
Multiply, then simplify
Solution:
12 x 3 y • 6 x 5 y 4 = 12 x 3 y • 6 x 5 y 4 = 72 x 8 y 5 = 36 • 2 • x 8 • y 4 • y = 6 x 4 y 2 2 y
Student Practice: Use the product rule to multiply, then simplify.
1.
50 • 2
2.
7• 7
3.
13 • 13
4.
3• 6
5.
2x 3 • 8x 3 y 4
6.
x3 • x2
7. 3 5 • 2 10
To multiply radical expressions with more than one term, use the product rule
discussed earlier along with the distributive property. Multiply the inside of the
radicals together and the outside of the radicals together, then simplify if possible.
Sample Problem:
Multiply
(2 3 − 5 2 )(3 3 + 2 )
Solution: (FOIL) ( 2 3 − 5 2 )(3 3 + 2 ) = 2 3 • 3 3 + 2 3 • 2 − 5 2 • 3 3 − 5 2 • 2
= 6 9 + 2 6 − 15 6 − 5 4
= 6 • 3 − 13 6 − 5 • 2
= 18 − 13 6 − 10
= 8 − 13 6
Student Practice:
1. 3 ( 5 + 2 )
Multiply and simplify.
3. (3 2 − a )(3 2 + a )
2. (1 − 7 )(4 + 3 7)
4. (2 5 − 3 2 )( 5 + 4 2 )
5. (5 + x ) 2
6. (a − b ) 2
• Dividing Radical Expressions
When dividing rational expressions, use the quotient rule mentioned before stating that
the quotient of two radicals is the radical of the quotient.
QUOTIENT RULE
a
b
Sample Problem: Divide and simplify.
Solution:
24 x 11
3x 7
=
=
a
b
24 x11
3x 7
24 x 11
= 8x 4 = 4 • 2 x 4 = 2 x 2 2
7
3x
Student Practice: Divide and simplify.
7.
75
3
8.
48 y 9
3y3
9.
24 x 7
10.
3x 2
18 x 5
3x
● RATIONALIZING DENOMINATORS
Often times in mathematics it is useful to write a fraction without a radical in the
denominator.
The process of writing a fraction with a radical in the denominator as an equivalent
fraction without a radical in the denominator is called rationalizing the denominator.
To rationalize a denominator, try the following:
• Multiply the numerator and denominator by a radical term that will make the
bottom radicand a perfect square.
Sample Problem:
Solution:
3
Rationalize
3
5
•
5
5
=
5
3 5
25
=
3 5
5
Student Practice: Rationalize each denominator.
10
11.
12.
3
3
7
13.
15.
5
14.
x
4
3 2
16.
3
8
2
3
15.5 Solving Radical Equations
To solve equations with radicals,
1. Isolate the radical on one side of the equation.
2. Raise each side to the power of the index of the radical.
If the equation still contains a radical, repeats steps 1 and 2.
3. Solve the resulting equation.
4. Ensure that your answer works in the original equation.
Sample Problem:
Solve for x
a.
x−5 +2 = 7
x−5 +2 = 7
Solution: a.
x−5 = 5
x − 5 = 25
x = 30
The solution is 30.
Solution: b.
2x + 1 + 1 = x
2x + 1 = x − 1
2 x + 1 = ( x − 1) 2
2x + 1 = x 2 − 2x + 1
0 = x 2 − 4x
0 = x ( x − 4)
x=0 x=4
Only x = 4 is a solution to the original equation.
b.
2x + 1 + 1 = x
30 − 5 + 2 = 7
Check:
25 + 2 = 7
5+2 = 7
Student Practice: Solve each equation for x.
1.
x + 11 = 15
2.
10 x − 1 − 6 = 1
3.
9 x + 10 + 5 = 15
4.
5x + 4 = x + 8
5.
24 + 2 x = x
7. 2 x + 1 + 6 = 3
6.
8 x + 32 − 4 = x