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 SurdA 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