Simplifying radical expressions, rational exponents, radical equations
Section 10.5 Expressions Containing Several Radical Terms.
-
Upload
dwayne-davidson -
Category
Documents
-
view
218 -
download
2
Transcript of Section 10.5 Expressions Containing Several Radical Terms.
Section 10.5
Expressions Containing Several Radical Terms
Definition
Like Radicals are radicals that have the same index and same radicand.
We can ONLY combine Like Radicals.
• 1) Simplify each radical.• 2) Combine like radicals.
To add/subtract radical expressions, we
Solution
Simplify by combining like radical terms.
3 3 32 2 2
a) 3 5 7 5
b) 7 9 9 2 9s s s
a) 3 5 7 5 (3 7) 5= 10 5
3 3 3 32 2 2 2b) 7 9 9 2 9 (7 1 2) 9s s s s
3 26 9s
Example
Solution
Simplify by combining like radical terms.
3 43
a) 2 18 7 2
b) 10m 3 24m m
a) 2 18 7 2 2 9 2 7 2
2(3) 2 7 2 6 2 7 2 2
3 43 3 3b) 10m 3 24 10m 3 2 3m m m m m 312 3m m
Example
ExamplesSimplify the following expressions
aaa 735
482122753
33 4 24 xxx 333 185032 xxx
254 3 xx
Product of two or more radical terms1. Use distributive law or FOIL
2. Use product rule for radicals
3. Simplify and combine like terms.
Examples: Multiply. Simplify if possible. Assume all variables are positive
nnn baab
xxn n
3 23
a) 2( 2)
b) 2 3
c)
y
x x
m n m n
Solution
a) 2( 2) 2 2 2y y
2 4 2 2y y
Using the distributive law
3 3 32 2 23 3 3b) 2 3 3 2 6x x x x x x
3 33 233 2 6x x x
3 233 2 6x x x
F O I L
Solution
c) m n m n
2 2m m n m n n
m n
F O I L
Notice that the two middle terms are opposites, and the result contains no radical. Pairs of radical terms like,
are called conjugate pairs.
and ,m n m n
Rationalizing Denominators with Two
Terms The sum and difference of the same terms are
called conjugate pairs.
To rationalize denominators with two terms, we multiply the numerator and denominator by the conjugate of the denominator.
Solution
Rationalize the denominator:3
5 2
5 2
5 2
3( 5 2)
25 10 10 4
3 3
5 2 5 2
Example
3( 5 2)
5 2
3( 5 2)
3
5 2
1
1
Solution
Rationalize the denominator:5
.7 y
7
7
y
y
2
5 7 5( 7 )
7 7 49 7 7
y y
y y y y y
25 7 5
7
y
y
5 5.
7 7y y
Example
Solution
Rationalize the denominator:4 m
m n
m n
m n
2
2 2
4 4m mn
m mn mn n
4 4.
m m
m n m n
Example
4 4m mn
m n
Terms with Differing Indices
To multiply or divide radical terms with different indices, we can convert to exponential notation, use the rules for exponents, and then convert back to radical notation.
Solution
Multiply and, if possible, simplify: 5 3 .x x
5 3 1/ 2 3/ 5x x x x 11/10x
10 11x
10 10 10x x 10x x
Converting to exponential notation
Adding exponents
Converting to radical notation
Simplifying
Example
Group ExerciseSimplify the following radical expressions
2( 3 2)x 6
7 4
24
5
2
a
a
233 32 3 4y y y