Rationalization of Surds: Formulas and Methods

Rationalization of Surds is the process of removing surds from the denominator of a fraction. It uses a suitable rationalising factor, usually the same surd for a single-term denominator or the conjugate for a binomial denominator. The method simplifies expressions without changing their values.

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What Is Rationalization of Surds?

Rationalization of surds means converting a fraction so that its denominator contains no irrational surd. This is done by multiplying the numerator and denominator by a suitable expression equal to 1.

For a denominator √a, multiply by √a. For a denominator √a + √b or √a − √b, multiply by its conjugate, √a − √b or √a + √b. The product of conjugates uses the difference of squares identity: (x + y)(x − y) = x² − y².

Rationalization of Surds Formula & Tricks

Important Formulas

Single square-root denominator
1/√a = √a/a, where a > 0

Multiply the numerator and denominator by √a because √a × √a = a, which is rational.

Numerical coefficient with a surd
p/(q√a) = p√a/(qa)

Multiply both parts of the fraction by √a and use (√a)² = a.

Conjugate of two-term surds
(√a + √b)(√a − √b) = a − b

The conjugate changes the sign between the two terms and removes the square roots from the product.

General binomial denominator
1/(p + q√r) = (p − q√r)/(p² − q²r)

Multiply by the conjugate p − q√r. The denominator follows (p + q√r)(p − q√r) = p² − q²r.

Cube-root denominator
1/∛a = ∛(a²)/a

Multiplying by ∛(a²) gives ∛(a³) = a in the denominator.

Quick Tricks

Use the conjugate for a binomial

For a denominator containing two terms joined by + or −, reverse the sign between the terms. This changes the denominator into a difference of squares.

Example: 1/(√5 + √2) = (√5 − √2)/(5 − 2) = (√5 − √2)/3.
Simplify square factors first

Take perfect-square factors outside the radical before rationalizing. This keeps the final expression shorter.

Example: 1/√12 = 1/(2√3) = √3/6.
Rationalize only the denominator

Multiply the numerator and denominator by the same rationalising factor. Then simplify the resulting numerator and denominator separately.

Example: 7/(2√3) = (7√3)/(2 × 3) = 7√3/6.

Rationalization of Surds Concepts

Rationalizing a Single Surd

A single surd in the denominator is removed by multiplying the fraction by that same surd.

For 1/√a, use √a/√a. The denominator becomes a rational number because √a × √a = a. If the numerator already contains a term, multiply that term by the same surd.

Example: 5/√7 = (5√7)/(√7 × √7) = 5√7/7.

Rationalizing with Conjugates

The conjugate of x + y is x − y, and the conjugate of x − y is x + y.

When the denominator has two surd terms, multiply by its conjugate. For √a + √b, the conjugate is √a − √b. Their product is a − b, so the denominator becomes rational.

Example: 3/(√7 − √3) = 3(√7 + √3)/(7 − 3) = 3(√7 + √3)/4.

Denominator of the Form p + q√r

For a denominator p + q√r, use the conjugate p − q√r.

Multiplication gives (p + q√r)(p − q√r) = p² − q²r. Thus, the rationalized form is obtained by placing the conjugate in the numerator and p² − q²r in the denominator.

Example: 1/(3 + 2√2) = (3 − 2√2)/(9 − 8) = 3 − 2√2.

Cube-Root Rationalization

For a single cube root in the denominator, multiply by the square of that cube root.

Since ∛a × ∛(a²) = ∛(a³) = a, the denominator becomes rational. For a denominator ∛a + ∛b, the identity x³ + y³ = (x + y)(x² − xy + y²) can be used.

Example: 1/∛5 = ∛25/∛125 = ∛25/5.

Rationalization of Surds Video Lessons

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13 Lessons
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Rationalising the Denominator in Surds

Learn how to rationalise denominators containing surds by applying appropriate algebraic methods and simplifying the resulting expressions accurately.

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Practice Rationalization of Surds Questions

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1Simplify the following expression for k = 2 and m = 1. (8^k × 18^m × 10^(k+m))/(4^(k+m) × 27^m × 5^(2k))→ 2(√20 + √45)/√5 = ?→ 3Simplify: 72 + √32 - √18→ 4If 5^((x - y)) = 125 and 5^((x + y)) = 3125, find x.→ 5Simplify: √(5 + √25 + 5² - 3² - 10) - √4→ 6Choose the incorrect relation from the following: I. √6 + √2 = √5 + √3 II. √6 + √2 √5 + √3→ 7A clerk calculates √45 + √20 .Simplify to surd form.→ 8An officer computes (√50 + √8).Simplify.→ 9If 4(x + 1) = 64 × 8x find the value of x.→ 10(√6+√2)+1/((√6-√2)) = ?→

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Quick Revision Notes

Rationalization of Surds: Quick Revision

Recall these formulas and rules before solving surd denominator expressions.

  • For 1/√a, multiply by √a; the result is √a/a.
  • For a binomial surd denominator, multiply by its conjugate.
  • The conjugate of p + q√r is p − q√r.
  • Use (x + y)(x − y) = x² − y².
  • For p + q√r, the conjugate product is p² − q²r.
  • Simplify perfect-square factors inside a radical before rationalizing.
  • For 1/∛a, multiply by ∛(a²) to obtain ∛(a³) = a.

Rationalization of Surds FAQs

What is the rationalising factor of √a?

The rationalising factor of √a is √a, because √a × √a = a.

What is the rationalising factor of √a + √b?

Its rationalising factor is √a − √b. Their product is a − b.

How do you rationalize 1/(√5 − √2)?

Multiply by the conjugate: 1/(√5 − √2) = (√5 + √2)/(5 − 2) = (√5 + √2)/3.

How do you rationalize 1/(2 + √3)?

Multiply by 2 − √3: 1/(2 + √3) = (2 − √3)/(4 − 3) = 2 − √3.

What is the rationalization formula for p + q√r?

1/(p + q√r) = (p − q√r)/(p² − q²r), provided p² − q²r is not zero.

How do you rationalize 1/√12?

First simplify √12 = 2√3. Then 1/√12 = 1/(2√3) = √3/6.

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