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Rationalizing the Denominator

Rationalizing the Denominator

The process by which a fraction is rewritten so that the denominator contains only rational numbers. A variety of techniques for rationalizing the denominator are demonstrated below.

 

Square Roots
(a > 0, b > 0, c > 0) 		Examples

Math equation showing step 1: a/√b = a/√b · √b/√b = a√b/b, demonstrating rationalization by multiplying by √b/√b.

 

Math example showing a/(b+√c) rationalized by multiplying by (b−√c)/(b−√c) to get (ab−a√c)/(b²−c)

Equation showing rationalization: 4/√2 = (4/√2)·(√2/√2) = 4√2/2 = 2√2

 

Step-by-step rationalization of 6/(3+√7) by multiplying by (3−√7)/(3−√7), yielding (18−6√7)/2 = 9−3√7.

 
Math example 3: a/(b−√c) = a/(b−√c) · (b+√c)/(b+√c) = (ab+a√c)/(b²−c)

Step-by-step rationalization of 5/(2−√3) by multiplying by (2+√3)/(2+√3), simplifying to 10+5√3.

 
Math example showing a/(√b + √c) rationalized by multiplying by (√b − √c)/(√b − √c) = (a√b − a√c)/(b − c)

Rationalizing 14/(√13+√11) by multiplying by (√13−√11)/(√13−√11), simplifying to (14√13−14√11)/2 = 7√13−7√11.

 
Math example showing: a/(√b−√c) = a(√b+√c)/(b−c), rationalizing by multiplying by (√b+√c)/(√b+√c).

Rationalizing 12/(√15−√7) by multiplying by (√15+√7)/(√15+√7), simplifying to (3√15+3√7)/2.

 

nth Roots
(a > 0, b > 0, c > 0) 		Examples

Rule 1: a divided by nth-root of b equals (a divided by nth-root of b) times (nth-root of b^(n-1) divided by nth-root of...

Simplification: 16/⁴√2 = (16·⁴√8)/2 = 8·⁴√8, achieved by rationalizing with ⁴√(2³) in numerator and denominator.

 
Rule 2: a divided by nth-root of b^m equals a divided by nth-root of b^m, multiplied by nth-root of b^(n-m) over nth-root of...

Simplifying 2/⁵√9: multiply by ⁵√3³/⁵√3³ to get 2⁵√27/3, showing rationalization of a fifth root denominator.

 

Rule 3: a/(b − ⁿ√c) = a(bⁿ⁻¹ + bⁿ⁻²ⁿ√c + … + ⁿ√cⁿ⁻¹) / (bⁿ − c), rationalizing

 

Example

Rationalizing 25/(3−∛2) by multiplying by (3²+3∛2+∛4)/same, yielding 25(9+3∛2+∛4)/25 = 9+3∛2+∛4

 

See also

Square root rules, nth root rules, irrational numbers, factoring rules