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Rationalizing the Denominator
Presented byKathryn Maloney
Binomial Denominators
Rationalizing the Denominator
(1) Multiply both the numerator and denominator by the conjugate of the denominator.
(2) Simplify the result.
Cool Math Term
(√𝒃)Conjugate Surd
A surd is a root that can’t be simplified further.
Product of Two Conjugates
(𝒂+√𝒃) (𝒂−√𝒃)
𝒂𝟐− (√𝒃 )𝟐
𝒂𝟐−𝒃
Product of the sum and difference of two terms.
𝒂𝟐−𝒂 √𝒃+𝒂√𝒃− (√𝒃)𝟐FOIL!
63+√5
Perform the indicate division. (Rationalize the denominator)
What is the conjugate of the denominator?
𝟑−√𝟓6
3+√5𝟑−√𝟓𝟑−√𝟓¿
6 (3−√5 )32−5
¿6 (3−√5 )
4¿3 (3−√5 )
2
12+√3
Perform the indicate division. (Rationalize the denominator)
You Try!
Press pause, find the solution, and play to check your answer!
12+√3
Perform the indicate division. (Rationalize the denominator)
What is the conjugate of the denominator?
𝟐−√𝟑1
2+√3𝟐−√𝟑𝟐−√𝟑¿
1 (2−√3 )22−3
¿(2−√3 )1
¿2−√3
You
Try
!
3−√24+√2
Perform the indicate division. (Rationalize the denominator)
What is the conjugate of the denominator?
4
3−√24+√2
𝟒−√𝟐𝟒−√𝟐
¿(3−√2 ) (4−√2 )
42−2
¿ 12−3√2−4 √2+214 ¿ 14−7 √2
14
¿7 (2−√2 )14
¿ 2−√22
FOIL
2+√31−√3
Perform the indicate division. (Rationalize the denominator)
You Try!
Press pause, find the solution, and play to check your answer!
2+√31−√3
Perform the indicate division. (Rationalize the denominator)
What is the conjugate of the denominator?
1
2+√31−√3
𝟏+√𝟑𝟏+√𝟑
¿(2+√3 ) (1+√3 )
12−3
¿ 2+2√3+√3+3−2 ¿− 5+3√32
You
Try
!
FOIL
3√2+12√2−1
Perform the indicate division. (Rationalize the denominator)
What is the conjugate of the denominator?
2√2+1
¿(3√2+1 ) (2√2+1 )
(2√2 )2−12
¿6(2)+3√2+2√2+1
8−1 ¿ 13+5√27
3√2+12√2−1
2√2+12√2+1
FOIL
2√2+33√2+2
Perform the indicate division. (Rationalize the denominator)
You Try!
Press pause, find the solution, and play to check your answer!
2√2+33√2+2
Perform the indicate division. (Rationalize the denominator)
What is the conjugate of the denominator?
3√2−2
¿(2√2+3 ) (3√2−2 )
(3√2 )2−22
¿6 (2 )−4√2+9√2−6
18−4 ¿ 6+5 √214
You
Try
!
2√2+33√2+2
3√2−23√2−2
FOIL
Thank you!
Don't worry about moving slowly, worry about standing still. -- Chinese proverb