What it does
A fraction like 5/(3 − √2) has an irrational number on the bottom. Rationalising the denominator rewrites it so the bottom is a whole number, which makes it easier to estimate, add to other fractions and compare. This page does it step by step for three kinds of fraction:
- k/√n: multiply top and bottom by √n, because √n × √n = n.
- k/(a + b√n): multiply top and bottom by the conjugate a − b√n, because (a + b√n)(a − b√n) = a² − b²n, a whole number.
- k/(√m ± √n): multiply by √m ∓ √n, because (√m + √n)(√m − √n) = m − n.
Each step is written with your numbers, the answer is simplified (common factors cancelled, and roots like √12 = 2√3 taken apart), and a decimal check shows the value has not changed. A picture shows why the conjugate works: a big square minus a small square is an L-shape that rearranges into a rectangle (x + y) by (x − y). It fits the Class 9 Maths chapter on the world of numbers (rationalising denominators) and the algebraic identities chapter.
The identity behind it is drawn with tiles in algebraic identity tiles; to see where a number like 2 − √3 sits on the line, zoom in with the number line zoom; and to construct √n with a ruler, use the square root spiral.
How to use it
- Pick the kind of fraction: k/√n, k/(a + b√n) or k/(√m ± √n).
- Type the whole numbers. b and a can be negative; n and m are the numbers under the roots.
- Read the steps, the answer, and the decimal check.
- If the bottom has no real root (for example 2 + √4 = 4) or is zero, the page says so.
A worked example
The page starts with 5/(3 − √2).
- Multiply top and bottom by (3 + √2)/(3 + √2), which is just 1.
- Bottom: (3 − √2)(3 + √2) = 3² − (√2)² = 9 − 2 = 7.
- Top: 5 × (3 + √2) = 15 + 5√2.
- So 5/(3 − √2) = (15 + 5√2)/7.
Check: 5/(3 − √2) ≈ 3.15301 and (15 + 5√2)/7 ≈ 3.15301.
Tips
- The conjugate of a + b√n is a − b√n: same numbers, opposite sign in the middle.
- When a² − b²n = 1, the fraction disappears completely: 1/(2 + √3) = 2 − √3.
- Simplify roots at the end: 3/√12 = 3√12/12 = 6√3/12 = √3/2.
Check yourself
Four fresh questions: k/√n as a√n, 1/(a + √b) as p + q√b, 1/(√(m + 1) + √m) as √A − √B, and the decimal value of a rationalised fraction. Type the numbers asked for, in order. Nothing is saved.
FAQ
Why bother? Before calculators, dividing by 1.414… by hand was slow; dividing by 7 was easy. Today it still makes answers easier to compare and to add, and examiners expect it.
What about cube roots? They need a different identity, a³ − b³ = (a − b)(a² + ab + b²), and are beyond this page.
Is this from the NCERT book? It covers the same topic with fibub’s own examples. fibub is not part of NCERT.
Where the numbers come from
- Standard school algebra: multiplying by √n/√n or by the conjugate, the identity (x + y)(x − y) = x² − y², and simplifying √(s²r) = s√r. Written for fibub; exact whole-number arithmetic.
- Checked against hand-worked examples (6/√3 = 2√3, 1/(2 + √3) = 2 − √3, 1/(√5 + √3) = (√5 − √3)/2, 5/(3 − √2) = (15 + 5√2)/7, 3/√12 = √3/2) and, for about 3,000 random fractions, the decimal value of the answer against that of the question.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.