chevron icon chevron icon chevron icon chevron icon

Surds

In this article, we will learn about

  • How to identify surds
  • Simplify expressions that involve surds

This article is specifically written to meet the requirements of Secondary 3 Additional Mathematics.

What is a Surd?

Surds are essentially irrational numbers that comprise the square root or cube root of a number.

Surds Not Surds
\(\sqrt{5}\) \(\sqrt{5}=25\)
\(\sqrt[3] {2}\) \(\sqrt[3] {8} =2\)
\(\begin{align*} \frac {1+\sqrt{2}}{5} \end{align*}\) \(\sqrt{4} \;+\; \sqrt[3]{27} = 5\)

In this article, we will focus only on surds that involve square roots like \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{10}\), \(\sqrt{11}\).

 

Question 1:

Identify the surd in the following

A. \(\begin{align*} \frac {\sqrt{100}+\sqrt{9}}{2} \end{align*}\)

B. \(\begin{align*} \frac {\sqrt{7}+\sqrt{4}}{5} \end{align*}\)

C. \(\begin{align*} \sqrt[3]{27}+1 \end{align*}\)

D. \(\begin{align*} 2\sqrt {4} \end{align*}\)

Solution:

B) \(\begin{align*} \frac {\sqrt{7}+\sqrt{4}}{5} \end{align*}\)

Explanation:

The option (B) is a surd because upon solving the equation, it gives an irrational value and a long string of Decimals.

Essentially, when you calculate the value of \(\sqrt{2}\) it is equal to \(1.4141414…..\)

In its Square root form, the quantity is called surd and the decimal form is called an irrational number.

Laws Governing Surds

Let us perform a few sets of exercises to understand how the simple operations around surds work.

      TRUE / FALSE
Addition  \(\begin{align*} \sqrt {5+3} = 2.828 \end{align*}\) \(\begin{align*} \sqrt {5}+\sqrt {3} = 3.968 \end{align*} \) FALSE
Subtraction \(\begin{align*} \sqrt {5-3} = 1.414 \end{align*}\) \(\begin{align*} \sqrt {5}-\sqrt {3} = 0.504 \end{align*} \) FALSE
Multiplication \(\begin{align*} \sqrt {5\times3} =3.872 \end{align*}\) \(\begin{align*} \sqrt {5}\times\sqrt {3} = 3.872 \end{align*} \) TRUE
Division \(\begin{align*} \sqrt {\frac{5}{3}} = 1.290 \end{align*}\) \(\begin{align*} \frac{\sqrt {5}}{\sqrt {3}} = 1.290 \end{align*} \) TRUE
Square \(\begin{align*} \sqrt {5^2} = 5 \end{align*}\) \(\begin{align*} (\sqrt {5})^2 = 5 \end{align*}\) TRUE

From this table, we observe that for addition and subtraction, it does not work but it works for multiplication, division, and square.

From the above exercise, we can now look at the law of surds.

If \(a > 0, b > 0\) then

Multiplication \(\begin{align*} \sqrt {a\times b} &= \sqrt {a} \,\times\, \sqrt {b} \end{align*}\)
Division \(\displaystyle{\sqrt {\frac {a}{b}} = \frac {\sqrt a}{\sqrt b}}\)
Square \(\sqrt {a} \,\times \sqrt {a} = (\sqrt a)^2 =a\)

Example 1:

Simplify each of the following

  1. \(\sqrt18\)
     
  2. \(\displaystyle{\sqrt 18 \over \sqrt 2}\)
     
  3. \(\sqrt2 \times \sqrt18\)

Solution:

1. \(\sqrt{18}\) can be written as \(\sqrt {6} \times \sqrt {3} \;\; \text{&}\;\; \sqrt {9} \times \sqrt {2}\)

However, in the first case \(\sqrt {6} \times \sqrt {3}\,,\) when we simplify they can only give an irrational form and not a simplified surd form.

The second case, however, \(\sqrt {9} \times \sqrt {2}\) can be simplified to,  \(3\sqrt {2}\) and hence, it is the simplest form available.
 

2. \(\begin{align}​​ \frac {\sqrt {18}}{\sqrt {2}} = \sqrt {\frac {18}{2}} = \sqrt {9} =3 \end{align}\) 

In this, we can unify the roots into one under the Law governing Surds. The further cancellation results in a perfect square which when under root gives the result 3.
 

3. \(\sqrt {2} \times \sqrt {18} = \sqrt {2\times 18} = \sqrt {36} =6\)

In this, we can unify the roots into one under the Law governing Surds. The further cancellation results in a perfect square which when under root gives the result 6.

 

Question 2:

Simplify \(\sqrt{12}\) without using a calculator.

Solution:

\(\sqrt {12} = \sqrt {3\times4} = \sqrt {3} \;\times\; \sqrt {4} = \sqrt {3} \;\times 2 = 2\sqrt{3}\)

 

Example 2:

Simplify \(\sqrt {50} + \sqrt {8}\) without using a calculator.

Solution:

\(\begin{align*} &\sqrt {50} + \sqrt {8}\\[2ex] &= \sqrt {2\times 25} + \sqrt {2\times 4} \\[2ex] &= \sqrt {2} \times \sqrt {25} + \sqrt {2} \times \sqrt {4}\\ &\quad \small{\text {( split up using the multiplication rule in Law of Surds)}} \\[2ex] &= \sqrt {2} \times 5 + \sqrt {2} \times 2 \\[2ex] &= 5\sqrt {2} + 2\sqrt {2} \\[2ex] &= 7\sqrt {2} \end{align*}\)

 

Question 3:

Simplify \(\sqrt {20} + \sqrt {45}\) without using a calculator.

Solution: 

\(\begin{align*} &\sqrt {20} + \sqrt {45}\\[2ex] &= \sqrt {4\times 5} + \sqrt {5\times 9}\\[2ex] &= \sqrt {4} \times \sqrt {5} + \sqrt {5} \times \sqrt {9}\\ &\quad \small{\text{( split up using the multiplication rule in Law of Surds)}} \\[2ex] &= 2\sqrt {5} + 3\sqrt {5} \\[2ex] &= 5\sqrt {5} \end{align*} \)

 

Example 3:

Simplify \((\,5-3\sqrt{2}\, )(\,1-\sqrt{2}\,)\) without using a calculator.

Solution: 

Using the Distributive method or the “Rainbow method”, we expand the brackets 

\(\begin{align*} & (\,5-3\sqrt{2}\, )(\,1-\sqrt{2}\,) \\[2ex] & = 5-5\sqrt {2} -3\sqrt{2} + (\,3\sqrt {2}\,)(\,\sqrt {2}\,)\\[2ex] & = 5 - 8\sqrt{2} +6 \\[2ex] & = 11 - 8\sqrt{2} \end{align*}\)

 

Question 3:

Simplify \((\,1+\sqrt{5} \;)(\,2-3\sqrt5\,)\) without using a calculator.

Solution: 

Using the Distributive method or the “Rainbow method”, we expand the brackets 

\(\begin{align*} &(\,1+\sqrt{5} \;)(\,2-3\sqrt5\,) \\ \\ &= 2-3\sqrt{5} +2\sqrt{5}-3 \times 5 \\ \\ &= 2-\sqrt{5}-15 \\ \\ &= -13-\sqrt{5} \end{align*}\)

Rationalising the Denominator of a Surd

A fraction with a surd in its denominator can be simplified by making the denominator a rational number.

You are supposed to rationalise a denominator by multiplying the numerator and denominator of a fraction with the same surd in the denominator.

 

Example 4:

Simplify the following by rationalising the denominator.

\(\displaystyle{\frac {5}{\sqrt{3}}}\)

Solution:

\(\begin{align*} \frac {5}{\sqrt{3}} &= \frac {5 \times \sqrt{3}} {\sqrt{3}\times \sqrt{3}} \\ \\ &= \frac {5\sqrt{3}}{3} \end{align*}\)

Conclusion

In this article, we have learnt how to identify Surds, which are defined as an irrational number that comprises the square root or the cube root. We have also learnt how to simplify expressions involving surds and how to split the composite number such that one of the numbers is a perfect square. All the topics we have discussed here are as per the Secondary 3 Additional Mathematics syllabus.

 


 

Continue Learning
Quadratic Functions in Real-World Context Equations and Inequalities
Logarithmic Functions Surds
Polynomials & Cubic Equations Partial Fraction
Exponential Functions Coordinate Geometry (Circles)
Linear Law Binomial Theorem
Resources - Academic Topics
icon expand icon collapse Primary
icon expand icon collapse Secondary
icon expand icon collapse
Book a free product demo
Suitable for primary & secondary
select dropdown icon
Our Education Consultants will get in touch with you to offer your child a complimentary Strength Analysis.
Book a free product demo
Suitable for primary & secondary
Claim your free demo today!
Claim your free demo today!
Arrow Down Arrow Down
Arrow Down Arrow Down
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
Geniebook CTA Illustration Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Geniebook CTA Illustration Geniebook CTA Illustration
Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Get a free diagnostic report of your child’s strengths & weaknesses!
Arrow Down Arrow Down
Arrow Down Arrow Down
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
Gain access to 300,000 questions aligned to MOE syllabus
Trusted by over 220,000 students.
Trusted by over 220,000 students.
Arrow Down Arrow Down
Arrow Down Arrow Down
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
media logo
Geniebook CTA Illustration
Geniebook CTA Illustration
Geniebook CTA Illustration
icon close
Default Wrong Input
Get instant access to
our educational content
Start practising and learning.
No Error
arrow down arrow down
No Error
*By submitting your phone number, we have
your permission to contact you regarding
Geniebook. See our Privacy Policy.
Success
Let’s get learning!
Download our educational
resources now.
icon close
Error
Error
Oops! Something went wrong.
Let’s refresh the page!