Section 10.5 Expressions Containing Several Radical Terms.
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Transcript of Section 10.5 Expressions Containing Several Radical Terms.
![Page 1: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/1.jpg)
Section 10.5
Expressions Containing Several Radical Terms
![Page 2: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/2.jpg)
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
![Page 3: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/3.jpg)
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
![Page 4: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/4.jpg)
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
![Page 5: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/5.jpg)
ExamplesSimplify the following expressions
aaa 735
482122753
33 4 24 xxx 333 185032 xxx
254 3 xx
![Page 6: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/6.jpg)
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
![Page 7: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/7.jpg)
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
![Page 8: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/8.jpg)
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
![Page 9: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/9.jpg)
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.
![Page 10: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/10.jpg)
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
![Page 11: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/11.jpg)
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
![Page 12: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/12.jpg)
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
![Page 13: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/13.jpg)
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.
![Page 14: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/14.jpg)
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
![Page 15: Section 10.5 Expressions Containing Several Radical Terms.](https://reader036.fdocuments.net/reader036/viewer/2022082517/56649edb5503460f94beb7f3/html5/thumbnails/15.jpg)
Group ExerciseSimplify the following radical expressions
2( 3 2)x 6
7 4
24
5
2
a
a
233 32 3 4y y y