To simplify a fraction with a radical in the denominator, you use a process called "rationalizing the denominator".
This eliminates the radical from the bottom of the fraction, which is considered an unsimplified form in mathematics. The method depends on whether the denominator has a single term or multiple terms.
How to simplify with a monomial radical
If the denominator is a monomial (a single term) with a radical, you can eliminate the radical by multiplying the numerator and denominator by that same radical. This works because multiplying a radical by itself removes the radical sign, leaving a rational number. Since you are multiplying the top and bottom of the fraction by the same value, you are essentially multiplying by one and not changing the fraction's overall value.
Example: Rationalizing a square root
To simplify 53the fraction with numerator 5 and denominator the square root of 3 end-root end-fraction
53√
, multiply both the numerator and denominator by 3the square root of 3 end-root
3√
:53⋅33=539=533the fraction with numerator 5 and denominator the square root of 3 end-root end-fraction center dot the fraction with numerator the square root of 3 end-root and denominator the square root of 3 end-root end-fraction equals the fraction with numerator 5 the square root of 3 end-root and denominator the square root of 9 end-root end-fraction equals the fraction with numerator 5 the square root of 3 end-root and denominator 3 end-fraction
53√⋅3√3√=53√9√=53√3
The resulting fraction has no radical in the denominator.
Example: Rationalizing a cube root
For higher-order radicals like 123the fraction with numerator 1 and denominator the cube root of 2 end-root end-fraction
123√
, you need to create a perfect cube in the denominator. Since 2⋅2⋅2=82 center dot 2 center dot 2 equals 8
2⋅2⋅2=8
, you need two more factors of 2 in the denominator's radical. Multiply the fraction by 223223the fraction with numerator the cube root of 2 squared end-root and denominator the cube root of 2 squared end-root end-fraction
223√223√
or 4343the fraction with numerator the cube root of 4 end-root and denominator the cube root of 4 end-root end-fraction
43√43√
:123⋅4343=4383=432the fraction with numerator 1 and denominator the cube root of 2 end-root end-fraction center dot the fraction with numerator the cube root of 4 end-root and denominator the cube root of 4 end-root end-fraction equals the fraction with numerator the cube root of 4 end-root and denominator the cube root of 8 end-root end-fraction equals the fraction with numerator the cube root of 4 end-root and denominator 2 end-fraction
123√⋅43√43√=43√83√=43√2
How to simplify with a binomial radical
If the denominator is a binomial with a radical, such as a+ba plus the square root of b end-root
𝑎+𝑏√
, multiply the numerator and denominator by the binomial's conjugate. The conjugate has the same terms but the opposite sign in the middle (e.g., the conjugate of 1+51 plus the square root of 5 end-root
1+5√
is 1−51 minus the square root of 5 end-root
1−5√
). This eliminates the radical terms when multiplying the binomials.
Example: Rationalizing a binomial denominator
To simplify 41+5the fraction with numerator 4 and denominator 1 plus the square root of 5 end-root end-fraction
41+5√
, multiply the fraction by the conjugate of the denominator, 1−51 minus the square root of 5 end-root
1−5√
, over itself:41+5⋅1−51−5the fraction with numerator 4 and denominator 1 plus the square root of 5 end-root end-fraction center dot the fraction with numerator 1 minus the square root of 5 end-root and denominator 1 minus the square root of 5 end-root end-fraction
41+5√⋅1−5√1−5√
Multiply the numerators and denominators:
-
Numerator: 4(1−5)=4−454 open paren 1 minus the square root of 5 end-root close paren equals 4 minus 4 the square root of 5 end-root
4(1−5√)=4−45√
-
Denominator: (1+5)(1−5)=12−(5)2=1−5=-4open paren 1 plus the square root of 5 end-root close paren open paren 1 minus the square root of 5 end-root close paren equals 1 squared minus open paren the square root of 5 end-root close paren squared equals 1 minus 5 equals negative 4
(1+5√)(1−5√)=12−(5√)2=1−5=−4
The fraction becomes 4−45-4the fraction with numerator 4 minus 4 the square root of 5 end-root and denominator negative 4 end-fraction
4−45√−4
. Simplify by dividing each term in the numerator by -4:4-4−45-4=-1+54 over negative 4 end-fraction minus the fraction with numerator 4 the square root of 5 end-root and denominator negative 4 end-fraction equals negative 1 plus the square root of 5 end-root
4−4−45√−4=−1+5√
Final simplification
After rationalizing the denominator, you may need to simplify further. Check if any radicals in the numerator can be simplified by finding perfect square factors. Also, look for common factors between the terms in the numerator and the denominator, remembering that terms inside a radical cannot be simplified with terms outside the radical.