To rationalize the denominator means:
Rewrite a fraction so there is NO radical (square root) in the denominator.
Example (not rationalized):
{1} / {sqrt{2}}
Rationalized:
{sqrt{2}} / {2}
Same value — just a different (preferred) form.
Why do we do this?
Historically (and in algebra classes), answers are written without radicals in the denominator because:
-
it’s easier to compare numbers
-
it simplifies later calculations
-
it avoids division by irrational numbers
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it’s the standard form teachers expect
So yes — this is something you’re expected to know.
Case 1: Denominator has ONE square root
Example:
{3} / {sqrt{5}}
Step-by-step:
Multiply top and bottom by sqrt{5}:
3/sqrt(5) * sqrt(5)/sqrt(5)
{3}{sqrt{5}} / 5 =
✔ Denominator is now rational.
Why does this work?
Because:
sqrt{5} * sqrt{5} = 5
We’re using the idea that multiplying by 1 (same top and bottom) doesn’t change the value.
Case 2: Denominator has a number AND a square root
Example:
{4} / {2*sqrt{3}}
Step 1: Multiply by sqrt{3}:
{4} / {2*sqrt{3}} * {sqrt{3}} / {sqrt{3}}
Step 2: Simplify:
= 4sqrt{3} / {2*3} = {2*sqrt{3}} / {3}
Case 3: Denominator has TWO terms (binomial)
This uses a conjugate.
What’s a conjugate?
For:
a + sqrt{b}
The conjugate is:
a – sqrt{b}
Example:
{5} / {2 + sqrt{3}}
Multiply top and bottom by the conjugate:
{5} / {2 + sqrt{3}} * {2 – sqrt{3}} / {2 – sqrt{3}}
Denominator:
(2 + sqrt{3})(2 – sqrt{3}) = 4 – 3 = 1
Final answer:
5(2 – sqrt{3}) = 10 – 5sqrt{3}
Important rule to remember
(a + b)(a – b) = a^2 – b^2
This is why conjugates work.