Teach Me Maths

Surds

How to simplify and combine surds, expand expressions exactly, and rationalise denominators without a calculator.

Keep roots exact

A surd is an irrational root, such as \sqrt3. Keep it exact when the question asks for an exact answer.

For non-negative a and b:

\sqrt{ab}=\sqrt a\sqrt b

This is a multiplication rule, not an addition rule.

Take out the largest square factor you can find. Remember that, for real a:

\sqrt{a^2}=|a|

The square-root symbol gives the non-negative root.

Worked example 1

Simplify \sqrt{147}

  1. 147=49\times3, and 49 is a square number.

  2. Your turn. What is \sqrt{49}?

    \sqrt{49}=7

  3. Answer: \sqrt{147}=7\sqrt3

Explore a square factor

Try it: compare k=2 and k=4. The side doubles while the area quadruples: \sqrt{2k^2}=k\sqrt2 for positive k

Combine only like surds

Simplify each root first. Terms with the same remaining root combine like algebraic terms:

a\sqrt d+b\sqrt d=(a+b)\sqrt d

Different roots cannot normally be combined into one term.

Worked example 2

Simplify \sqrt{75}+\sqrt{48}-\sqrt{12}

  1. \sqrt{75}=5\sqrt3, \sqrt{48}=4\sqrt3 and \sqrt{12}=2\sqrt3

  2. Your turn. Combine the coefficients: 5+4-2

    5+4-2=7

  3. Answer: 7\sqrt3

Worked example 3

Expand and simplify (2+\sqrt7)(5-\sqrt7)

  1. Multiply all four pairs: 10-2\sqrt7+5\sqrt7-(\sqrt7)^2

  2. Your turn. What is (\sqrt7)^2?

    (\sqrt7)^2=7

  3. Answer: 10-7+3\sqrt7=3+3\sqrt7

Rationalise the denominator

Rationalising means writing an equivalent fraction with no surd in its denominator. Multiply the numerator and denominator by the same non-zero expression, so the value does not change.

For a denominator a+\sqrt b, use its conjugate a-\sqrt b. Their product is a^2-b, because the two cross terms cancel.

Worked example 4

Rationalise and simplify \frac{10}{\sqrt6}

  1. Multiply top and bottom by \sqrt6: \frac{10\sqrt6}{6}

  2. Your turn. Simplify \frac{10}{6}

    \frac{10}{6}=\frac53

  3. Answer: \frac{5\sqrt6}{3}

Worked example 5

Rationalise and simplify \frac{5}{3+\sqrt2}

  1. Use the conjugate 3-\sqrt2: \frac{5(3-\sqrt2)}{(3+\sqrt2)(3-\sqrt2)}

  2. Your turn. Work out the new denominator, 3^2-(\sqrt2)^2

    9-2=7

  3. Answer: \frac{15-5\sqrt2}{7}

Worked example 6

A rectangle has area 8+3\sqrt5 square units and width 3+\sqrt5 units. Find its length in the form a+b\sqrt5.

  1. Length is area divided by width: \frac{8+3\sqrt5}{3+\sqrt5}
  2. Multiply by the conjugate. The numerator is (8+3\sqrt5)(3-\sqrt5)=24-8\sqrt5+9\sqrt5-15

  3. Your turn. What is the rational part of this numerator, 24-15?

    The numerator is 9+\sqrt5

  4. The denominator is 3^2-5=4
  5. Answer: \frac94+\frac14\sqrt5 units. Multiplying by the original width gives the original area.

Common mistakes

  • Splitting a square root over addition. Square roots do not distribute over addition; simplify inside the root first.
  • Combining unlike surds. Simplify first, then collect only matching surd parts.
  • Missing products when expanding. Multiply every term in one bracket by every term in the other.
  • Changing only the denominator. When rationalising, multiply the numerator and denominator by the same expression.
  • Using the wrong conjugate. Keep the terms and reverse the sign between them.
  • Rounding an exact answer. Leave the answer in simplified surd form when exactness is requested.

Now try it: Surds practice questions

More on this topic: Surds worksheet with full solutions