Rationalizing a Denominator

Rationalizing a denominator means multiplying a fraction by a disguised 1 so the radical leaves the denominator.

Rationalize 4/√7. Multiply by √7/√7 — a fraction equal to 1, so the value is untouched: 4/√7 · √7/√7 = 4√7 / 7. The radical now lives upstairs. The same move clears any single-term radical denominator: 5/(2√3) times √3/√3 gives 5√3/6. Check one numerically: 4/√7 ≈ 1.5119 and 4√7/7 ≈ 1.5119 — the same number in a new dress. The written form changed; the value did not, because multiplying by 1 changes nothing. The name says why the move matters: the finished denominator is a rational number, 7 or 6, with no radical left in it — a form that adds and compares cleanly.