Math 51 – Prof. Beydler
8.4/8.5 – Notes
Page 1 of 4
Rationalizing the Denominator, and More Simplifying
Part of simplifying an expression is making sure there are no radicals on the denominators. This is
called rationalizing the denominator.
Ex 1.
Simplify by rationalizing the denominator.
9
√6
Ex 2.
Simplify by rationalizing the denominator.
12
√8
Ex 3.
Simplify.
√
27
5
Ex 4.
Simplify.
5
1
√ ⋅√
8
6
Ex 5.
Simplify.
√4𝑥
√𝑦
Math 51
Ex 6.
Simplify.
√
2𝑥 2 𝑦
3
Ex 7.
Simplify.
3
√
3
2
Ex 8.
Simplify.
3
3
√2
√3𝑥2
Ex 9.
Simplify.
(√3 + √21)(√3 − √7)
Ex 10.
Simplify.
(4 + √3)(4 − √3)
8.4/8.5 – Notes
Page 2 of 4
8.4/8.5 – Notes
Math 51
Ex 11.
Simplify.
6+√2
√2−5
Ex 12.
Simplify.
4
3+√𝑥
When simplifying with radicals:
1. Use the product rule to bring radicals together. (ex: √2 ⋅ √𝑥 = √2𝑥)
2. Pull as much as possible out of the radicals. (ex: √8 = √4 ⋅ 2 = 2√2)
3. Combine like terms. (ex: 3√2 + 4√2 = 7√2)
4. Rationalize denominators. (ex:
7
√3
=
7⋅√3
√3⋅√3
=
7√3
3
)
Practice
1. Simplify.
a)
3
√24
5
b) √18
5𝑥 3
c) √
𝑦
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8.4/8.5 – Notes
Math 51
3
d)
√3
3
√4
e) (√10 − 7)
2
5
f) 3+√5
Q: Suppose your boyfriend/girlfriend sends you this text message:
A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,V,W,X,Y,Z.
What does the message mean?
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