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Algebra

Surds

Introduction

Surds are numbers left in root form \((\surd)\) to express their exact value. They have an infinite number of non-recurring decimals. Therefore, surds are irrational numbers.

There are certain rules that we follow to simplify an expression involving surds. Rationalising the denominator is one way to simplify these expressions. It is done by eliminating the surd in the denominator. This is shown in rules 5 and 6.

These are the following rules that we follow to simplify an expression involving surds.

Rules for simplifying surds

Rule 1

\[ \sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b} \]

Simplify \(\sqrt{18}\)

Question

Simplify \(\sqrt{18}\)

Solution
  1. Step 1 — Find the largest perfect square factor

    Since \(18 = 9 \times 2 = 3^{2} \times 2\), as 9 is the largest perfect square factor of 18:

    \[ \therefore\ \sqrt{18} = \sqrt{3^{2} \times 2} \]
  2. Step 2 — Split the surd

    Using the rule \(\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}\):

    \[ = \sqrt{3^{2}} \times \sqrt{2} \]
  3. Step 3 — Evaluate the perfect square

    The square root of \(3^{2}\) is 3, so the simplified surd is three times the square root of two:

    \[ \sqrt{18} = 3\sqrt{2} \]

Rule 2

\[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]

Simplify \(\sqrt{\frac{12}{121}}\)

Question

Simplify \(\sqrt{\dfrac{12}{121}}\)

Solution
  1. Step 1 — Split the root over the fraction

    Using the rule \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\):

    \[ \sqrt{\frac{12}{121}} = \frac{\sqrt{12}}{\sqrt{121}} \]
  2. Step 2 — Factorise inside the root

    Since 4 is the largest perfect square factor of 12, and \(\sqrt{121} = 11\):

    \[ = \frac{\sqrt{2^{2} \times 3}}{11} \]
  3. Step 3 — Split the surd

    Using the rule \(\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}\):

    \[ = \frac{\sqrt{2^{2}} \times \sqrt{3}}{11} \]
  4. Step 4 — Evaluate the perfect square

    The square root of \(2^{2}\) is 2, so the simplified surd is two root three over eleven:

    \[ \sqrt{\frac{12}{121}} = \frac{2\sqrt{3}}{11} \]

Rule 3

\[ \frac{b}{\sqrt{a}} = \frac{b}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{b\sqrt{a}}{a} \]

Rationalise \(\frac{5}{\sqrt{7}}\)

Question

Rationalise \(\dfrac{5}{\sqrt{7}}\)

Solution
  1. Step 1 — Multiply both numerator and denominator by \(\sqrt{7}\)
    \[ \frac{5}{\sqrt{7}} = \frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} \]
  2. Step 2 — Simplify

    The denominator becomes the whole number 7, so the rationalised fraction is five root seven over seven:

    \[ \frac{5}{\sqrt{7}} = \frac{5\sqrt{7}}{7} \]

Rule 4

\[ a\sqrt{c} \pm b\sqrt{c} = (a \pm b)\sqrt{c} \]

Simplify \(5\sqrt{6} + 4\sqrt{6}\)

Question

Simplify \(5\sqrt{6} + 4\sqrt{6}\)

Solution
  1. Step 1 — Collect the like surds

    Using the rule \(a\sqrt{c} \pm b\sqrt{c} = (a \pm b)\sqrt{c}\):

    \[ 5\sqrt{6} + 4\sqrt{6} = (5 + 4)\sqrt{6} \]
  2. Step 2 — Add

    The simplified surd is nine times the square root of six:

    \[ 5\sqrt{6} + 4\sqrt{6} = 9\sqrt{6} \]

Rule 5

\[ \frac{c}{a + b\sqrt{n}} \]

For a fraction of this form, multiply top and bottom by \(a - b\sqrt{n}\).

Rationalise \(\frac{3}{2 + \sqrt{2}}\)

Question

Rationalise \(\dfrac{3}{2 + \sqrt{2}}\)

Solution
  1. Step 1 — Multiply the numerator and denominator by \(2 - \sqrt{2}\)
    \[ \frac{3}{2 + \sqrt{2}} = \frac{3}{2 + \sqrt{2}} \times \frac{2 - \sqrt{2}}{2 - \sqrt{2}} \]
  2. Step 2 — Expand the numerator and denominator
    \[ = \frac{6 - 3\sqrt{2}}{4 - 2} \]
  3. Step 3 — Simplify the denominator

    The rationalised fraction is six minus three root two, all over two:

    \[ \frac{3}{2 + \sqrt{2}} = \frac{6 - 3\sqrt{2}}{2} \]

Rule 6

\[ \frac{c}{a - b\sqrt{n}} \]

For a fraction of this form, multiply top and bottom by \(a + b\sqrt{n}\).

Rationalise \(\frac{3}{2 - \sqrt{2}}\)

Question

Rationalise \(\dfrac{3}{2 - \sqrt{2}}\)

Solution
  1. Step 1 — Multiply the numerator and denominator by \(2 + \sqrt{2}\)
    \[ \frac{3}{2 - \sqrt{2}} = \frac{3}{2 - \sqrt{2}} \times \frac{2 + \sqrt{2}}{2 + \sqrt{2}} \]
  2. Step 2 — Expand the numerator and denominator
    \[ = \frac{6 + 3\sqrt{2}}{4 - 2} \]
  3. Step 3 — Simplify the denominator

    The rationalised fraction is six plus three root two, all over two:

    \[ \frac{3}{2 - \sqrt{2}} = \frac{6 + 3\sqrt{2}}{2} \]

You are now ready to try this topic's questions. Go to Question 1