Surds

Surds

In this article, we look at surds, simplification of surds, arithmetic of surds, and many more.

Douglas Tawiah Dwumor
February 25, 2024
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  • Reducing Surds to basic form
  • Perfect Squares
  • simplifying Surds

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Introduction

A surd is a positive number under the square root that cannot be further simplified into a rational number. Surds are irrational numbers. That’s, surds cannot be expressed as a number in the form \frac{a}{b}, b \neq 0. Examples of surds are \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7} etc.

Reducing Surds to basic form

A surd is said to be in basic form if it cannot be simplified further.

Examples of surds in basic form include \sqrt{3}, \sqrt{5}, \sqrt{7}, \sqrt{19}  etc

To simplify a surd is to reduce it to its basic form. If you can express the number under the square root as a product of a perfect square and another number, then the surd is not in basic form.

For example, \sqrt{27} is not in basic form because 27 can be expressed as product of a perfect square factor and another factor. That’s, \sqrt{27}=\sqrt{9 \times 3}. 

Before we proceed, we advise that the student notes these perfect square numbers.

Number Perfect Square
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
14 196
15 225
16 256
17 289
18 324
19 361
20 400

Worked Examples

Simplify the following:

(i) \sqrt{8}

(ii) \sqrt{48}

(iii) \sqrt{98}

(iv) \sqrt{80}

Solution

(i) \sqrt{8}

write 8 as product of perfect square and a number that can’t be simplified further

= \sqrt{4 \times 2}

Re-write it as product of two surds

= \sqrt{4 } \times \sqrt{ 2}

simplify the perfect square under the square root and leave the other surd in basic form

= 2 \times \sqrt{2}

multiply to get final term

= 2 \sqrt{2}

(ii) \sqrt{48}

write 48 as product of perfect square and a number that can’t be simplified further

= \sqrt{16 \times 3}

Re-write it as product of two surds

= \sqrt{16 } \times \sqrt{ 3}

simplify the perfect square under the square root and leave the other surd in basic form

= 4 \times \sqrt{3}

multiply to get final term

= 4\sqrt{3}

(iii) \sqrt{98}

write 98 as product of perfect square and a number that can’t be simplified further

= \sqrt{49 \times 2}

Re-write it as product of two surds

= \sqrt{49 } \times \sqrt{ 2}

simplify the perfect square under the square root and leave the other surd in basic form

= 7 \times \sqrt{2}

multiply to get final term.

= 7\sqrt{2}

 (iv) \sqrt{80}

write 80 as product of perfect square and a number that can’t be simplified further

= \sqrt{16 \times 5}

Re-write it as product of two surds

= \sqrt{16 } \times \sqrt{ 5}

simplify the perfect square under the square root and leave the other surd in basic form

= 4 \times \sqrt{5}

multiply to get final term

= 4\sqrt{5}

Finding the right perfect square in the number under the square root

When resolving a surd into its basic form, the student’s daunting task is to find out the greatest perfect square number that can divide the number under the square root. For example, given \sqrt{32}, a student may quickly identify that 4, a perfect square, is a factor of 32 and may express \sqrt{32} as \sqrt{32} = \sqrt{4 \times 8}.

Let’s explore the sequence of solution step-by-step:

\sqrt{32} = \sqrt{4 \times 8}

4 is a perfect square alright. However, 8 still can be expressed as a product of a perfect square ,4, and 2. This makes the process daunting

=\sqrt{4} \times \sqrt{8}

=2 \times \sqrt{8}

=2 \times \sqrt{4 \times 2}

=2 \times \sqrt{4} \times \sqrt{ 2}

=2 \times 2 \times \sqrt{ 2}

=4 \sqrt{ 2}

To make the process simple and easy, we use the greatest perfect number that can divide 32. This is, 16. 

That’s, \sqrt{32}=\sqrt{16} \times \sqrt{2}

=\sqrt{16} \times \sqrt{2}

 =4\times \sqrt{2}

 =4\sqrt{2}        

Let’s consider two ways the student may use to find the greatest perfect square that can divide the number under the square root.

(1) Continuously expressing the number under the square root as product of factors until you get product of numbers that do not have perfect squares as factors.

For example, \sqrt{32}=\sqrt{4 \times 8} (we realize 8 contains another perfect square,4, so we again express 8 as 4×2

=\sqrt{4 \times 4 \times 2}

we now can form another perfect square from multiplying a perfect square by itself. That’s, 4×4 gives us a greater perfect square, 16

Hence, \sqrt{32}=\sqrt{16 \times 2}

We can now proceed to simplify.

(2) Using the scientific calculator.

If the student is permitted to use a calculator which has the square root function, it’s relatively easier to fetch the greatest perfect square factor.

For example, given \sqrt{32},

Step 1: Type \sqrt{32} on the calculator and press the Equal Sign Key. The output is 4\sqrt{2}.

Step 2: Square the output, 4\sqrt{2}, to get the product of the two factors. That’s, \left(4\sqrt{2}\right)^2= \left(4 \times \sqrt{2}\right)^2= \sqrt{16 \times 2} . Note that \left(\sqrt{2}\right)^2=2. That’s, if you square a surd, you get a rational number.

Therefore, \sqrt{32}=\sqrt{16} \times \sqrt{2}

Illustrative Examples

Example 1: Simplify the following:

i) \sqrt{125}     ii) \sqrt{162}    iii) \sqrt{96}   iv) \sqrt{75}   v) \sqrt{18}   vi) \sqrt{200}   vii) \sqrt{147}   viii) \sqrt{294}   ix) \sqrt{216}   x) \sqrt{32}  

Solution

i) \sqrt{125}=\sqrt{25 \times 5}

=\sqrt{25} \times \sqrt{5}}

=5 \times \sqrt{5}}

=5 \sqrt{5}}

iii) \sqrt{96}=\sqrt{16 \times 6}

=\sqrt{16} \times \sqrt{6}

=4 \times \sqrt{6}

=4\sqrt{6}

v) \sqrt{18}=\sqrt{9 \times 2}

=\sqrt{9} \times \sqrt{2}

=3 \times \sqrt{2}

=3\sqrt{2}

vii) \sqrt{147}=\sqrt{49 \times 3}

=\sqrt{49} \times \sqrt{3}

=7 \times \sqrt{3}

=7\sqrt{3}

ix) \sqrt{216}=\sqrt{36 \times 6}

=\sqrt{36} \times \sqrt{6}

=6 \times \sqrt{6}

=6\sqrt{6}

ii) \sqrt{162}=\sqrt{81 \times 2}

=\sqrt{81} \times \sqrt{2}

=9 \times \sqrt{2}

=9\sqrt{2}

iv) \sqrt{75}=\sqrt{25 \times 3}

=\sqrt{25} \times \sqrt{3}

=5 \times \sqrt{3}

=5\sqrt{3}

vi) \sqrt{200}=\sqrt{100 \times 2}

=\sqrt{100} \times \sqrt{2}

=10 \times \sqrt{2}

=10\sqrt{2}

viii) \sqrt{294}=\sqrt{49 \times 6}

=\sqrt{49} \times \sqrt{6}

=7 \times \sqrt{6}

=7\sqrt{6}

x) \sqrt{32}=\sqrt{16 \times 2}

=\sqrt{16} \times \sqrt{2}

=4 \times \sqrt{2}

=4\sqrt{2}

Addition and Subtraction of Surds

Adding two rational numbers is simple and straightforward. For example, 5 + 3 =8.

Also, subtracting a rational number from another rational number is simple. For example,5-3=2.

Adding two surds or irrational numbers, however,  can only be done if the surds are alike, or if they are like-terms. For example, \sqrt{5} + \sqrt{5} = 2\sqrt{5}.  That’s, one \sqrt{5} plus one \sqrt{5} gives twice of \sqrt{5}.

Note that, \sqrt{5} + \sqrt{5} \neq \sqrt{10}. (you cannot just add the rational numbers under the square root signs. You need to consider the entire \sqrt{5} as one number, an irrational number or surd).

Another example, is 4\sqrt{3} + 7\sqrt{3} = 11\sqrt{13}. Here, 4 of \sqrt{3} plus 7 of \sqrt{3} gives 11 of the \sqrt{3}.

Once again, let’s state the 4\sqrt{3} + 7\sqrt{3} = \neq 11\sqrt{6}.

It is safe to consider the square root part of the term as a variable. For example, given 4\sqrt{3} + 7\sqrt{3}, if we assume that \sqrt{3}=x, then we can write

4\sqrt{3} + 7\sqrt{3}=4x + 7x

=11x

but x=\sqrt{3}

Thus, we have 11\sqrt{3}.

Rules for adding or subtracting non-alike surds

Two surds that have different terms under the square root are not alike. For example, 2\sqrt{5} and 5\sqrt{2}  are not alike because the terms under the square root are not the same.

(1) m\sqrt{b} + n\sqrt{c} = m\sqrt{b} + n\sqrt{c}

(2) m\sqrt{b} - n\sqrt{c} = m\sqrt{b} - n\sqrt{c}

In the above, we cannot add or subtract the two surds because they are not alike or like-terms. Note that m\sqrt{b} and n\sqrt{c} are all surds but because the terms under the square roots are not the same, they are not like-terms.

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