Dividing Radicals and Rationalizing Denominators
Simple Explanation
To divide two radicals with the same index, divide the radicands under one radical: ⁿ√a / ⁿ√b = ⁿ√(a/b). When a radical is left in the denominator of a fraction, "rationalize" it by multiplying top and bottom by that radical, which removes it from the denominator.
Why Do We Need It?
A radical left in a denominator is considered an unsimplified, awkward form — rationalizing is the standard way to write a final answer cleanly, and is required before comparing or combining fractions that involve roots.
Formula
Quotient Rule for Radicals
ⁿ√a / ⁿ√b = ⁿ√(a/b) (b ≠ 0)
The quotient of two radicals with the same index equals the radical of the quotient of their radicands.
- a, b
- — the radicands (a ≥ 0, b > 0 when n is even)
- n
- — the common index (root) of both radicals
When to use it: Whenever you divide two radicals with the same index, including rationalizing a denominator that contains a radical.
Worked Example
Rationalize a denominator
Rationalize 5/√3.
Why Does This Work?
Multiplying by √3/√3 is multiplying by 1, so the value of the fraction never changes — but √3 · √3 = 3 removes the radical from the denominator, leaving an equivalent fraction in the accepted simplified form.
Real-Life Example
Physics formulas with a root in the denominator
A pendulum period formula can produce an intermediate expression with √g in the denominator.
Rationalizing before plugging in numbers avoids dividing by an irrational decimal approximation, keeping the calculation exact for longer.
Practice
Rationalize 4/√2.
MediumSimplify √20 / √5.
HardCommon mistake
Multiplying only the denominator by the rationalizing radical and forgetting to multiply the numerator too — you must multiply the whole fraction by (radical)/(radical) so the value stays unchanged.
Quick Review
- ⁿ√a / ⁿ√b = ⁿ√(a/b) — divide the radicands under one radical.
- Rationalize a denominator by multiplying top and bottom by the radical in the denominator.
- A fully simplified radical expression never has a radical left in the denominator.