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