Surds and rationalizing the denominator

Roots that never come out

A surd is an irrational radical with a rational radicand. √5 is a surd because no whole number squares to 5. √9 is not, since it simplifies to 3.

√π and √e are irrational but not surds, because π and e are not rational. Your book: every surd is irrational, but not every irrational number is a surd.

A monomial surd has one term, like √5. A binomial surd is a sum of two, like √3 + √5. √a + √b and √a − √b are conjugates, and their product is rational.

To rationalize a denominator, multiply top and bottom by its conjugate. The bottom becomes (√a)² − (√b)² = a − b, with no root left.

Change only the sign between the two terms to get the conjugate. Then (√a + √b)(√a − √b) = a − b.

Key terms

Surd

Roots that stay as roots.

Binomial surd

A sum of two monomial surds.

Conjugate surds

Their product has no root.

Rationalization

Clearing roots from a denominator.

Short questions with model answers

1. Q1. Example 8 (ii): rationalize 3/(√5 − √3).

   • 3/(√5 − √3) × (√5 + √3)/(√5 + √3)
   • 3(√5 + √3)/((√5)² − (√3)²) = 3(√5 + √3)/(5 − 3)

   3(√5 + √3)/2.

2. Q2. Is √9 a surd? Is √π?

   • √9 = 3
   • π is not rational

   Neither is a surd (p. 12).

Common mistakes

• ✗ Multiplying only the denominator by the conjugate.

  ✓ Multiply top and bottom, so the value stays the same.

• ✗ Using √5 + √2 itself to rationalize 3/(√5 + √2).

  ✓ (√5 + √2)² = 7 + 2√10 still has a root. Use √5 − √2.

• ✗ Calling √π a surd because it is irrational.

  ✓ The radicand π is not rational, so √π is not a surd (p. 12).

MCQs

1. 1. Which one is a surd?

   1. (a) √9
   2. (b) √π
   3. (c) √7
   4. (d) √e
   Show answer

   (c) 7 is rational and √7 is irrational (p. 12).

2. 2. The conjugate of √5 − √3 is:

   1. (a) √5 + √3
   2. (b) √3 − √5
   3. (c) −√5 − √3
   4. (d) √5 − √3
   Show answer

   (a) Change only the middle sign (p. 13).

3. 3. (√5 + √2)(√5 − √2) =

   1. (a) 7
   2. (b) √3
   3. (c) 3
   4. (d) 10
   Show answer

   (c) 5 − 2 = 3, a rational number.

4. 4. 3/(√5 − √3) rationalized is:

   1. (a) √5 + √3
   2. (b) 3(√5 − √3)/2
   3. (c) (√5 + √3)/2
   4. (d) 3(√5 + √3)/2
   Show answer

   (d) 5 − 3 = 2 stays in the bottom (Example 8 ii).

5. 5. (3 + √5)(3 − √5) is:

   1. (a) an irrational number
   2. (b) a rational number
   3. (c) a surd
   4. (d) undefined
   Show answer

   (b) 9 − 5 = 4; conjugates multiply to a rational (Review Exercise 1 x).

Quick revision

• A surd is an irrational root of a rational number; √π is not one.
• Conjugates √a ± √b multiply to a − b, a rational number.
• Rationalize by multiplying top and bottom by the conjugate.