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The Latin meaning of the word "Surd" is deaf or mute.  In earlier days, Arabian mathematicians called rational numbers and irrational numbers as audible and inaudible. Since surds form are made of irrational numbers, they were referred to as asamm (deaf, dumb) in Arabic language, and were later translated in Latin as surds.

With this awareness, let's move ahead to learn the working of surds in math.

In this mini-lesson, we shall explore the topic of surds,   by finding answers to questions like what do you mean by surds, what are the rules of surds, and simplifying calculations with surds.

Lesson Plan

What do you mean by surds.

Surd is simply used to refer to a number that does not have a root. 

\(\sqrt 4 \), \(\sqrt[3] 8 \), \(\sqrt 25 \) have roots as answers.

But \(\sqrt 6 \), \(\sqrt[3]2 \), \(\sqrt20 \) do not have proper roots.

These number forms are termed as surds.

Also, these surds example expressed in exponential form  would have fractions as powers.

\[\begin{align}\sqrt 6 &= 6^{\frac{1}{2}} \\ \sqrt[3]2 &= 2^{\frac{1}{3}} \end{align} \]

According to surds definition, there are three different types of surds.

Pure Surds: A surd having only a single irrational number is called a pure surd. \[\sqrt7,~ \sqrt[4]11,~ \sqrt x^3 \]

Mixed Surds: A surd having a mix of a rational number and an irrational number is called a mixed surd. \[x\sqrt y,~ 4\sqrt3,~ 8\sqrt5 \]

Compound Surds:   A surd composed of two surds is called a compound surd. \[4 + \sqrt3,~ \sqrt5 + \sqrt2,,~\sqrt a + b\sqrt c \]

What Are the Rules of Surds?

A few rules of working with surds are:

• Surds cannot be added.  \[\sqrt a + \sqrt b \neq \sqrt(a + b) \]
• Surds cannot be subtracted.\[\sqrt a - \sqrt b \neq \sqrt(a - b) \]
• Surds can be multiplied. \[\sqrt a \times \sqrt b = \sqrt(a \times b) \]
• Surds can be divided. \[ \frac{\sqrt a}{\sqrt b} = \sqrt\frac{a}{b} \]
• Surds can be written in exponential form.   \[\begin{align} \sqrt a &= a^{\frac{1}{2}} \\  \sqrt[n] a&= a^{\frac{1}{n}}\end{align}\]

Generally, surds cannot be added. But we can add similar surds. \[ \begin{align}m\sqrt a + n\sqrt a &= (m +n)\sqrt a  \\ 3\sqrt5 + 2\sqrt 5 &= 5\sqrt 5\end{align}\]

Now try to find the solution for the following surds. \[ 2\sqrt5 + \sqrt3 + \sqrt 2 + \sqrt 5 + 2\sqrt3 = ?\] 

Simplify Surd Calculation with Steps

Simplification of surds is needed for performing calculations.

There are two simple steps to surd simplification.

STEP - 1:   Split the number within the root into its prime factors .  \[\sqrt50 = \sqrt(5 \times 5 \times 2) \]

STEP-II:   Based on the root write the prime factors, outside the root. In case of square root, write one factor outside the root, for every two similar factors within the root.  \[\sqrt(5 \times 5 \times 2) = 5\sqrt 2 \]

\[\sqrt 18 + \sqrt50  \]

The above operation of addition is not possible without further simplification.

\[\begin{align}\sqrt18  + \sqrt50&= \sqrt (3 \times 3 \times 2)  + \sqrt(5 \times 5 \times 2) \\ &= 3\sqrt2  + 5\sqrt2 \\ &=8\sqrt2\end{align} \]

Solved Examples

Convert \(\sqrt 80 \) into mixed surd.

\[\begin{align}\sqrt 80 &=\sqrt (16 \times 5) \\ &= \sqrt(2 \times 2 \times 2 \times 2 \times 5 )\\ &= 2 \times 2 \times \sqrt 5 \\ &= 4\sqrt 5 \end{align} \]

Find the product of \(4 \sqrt 3 \) and \(2 \sqrt 5 \)

\[\begin{align} 4 \sqrt 3  \times 2 \sqrt 5  &= 4 \times 2 \times \sqrt3 \times \sqrt 5\\ &=8 \times \sqrt(3 \times 5)  \\ &= 8\sqrt 15\end{align} \]

Write the expansion of \((4 + 3\sqrt 2)^2 \)

\[\begin{align} (4 + 3\sqrt 2)^2  &= 4^2 + (3 \sqrt 2)^2 + 2.4.3\sqrt 2\\ &= 16 + 18 + 24\sqrt2 \\ &= 34 + 24\sqrt2\end{align} \]

Find the sum of  \((2\sqrt 5 - 4\sqrt 2)\) , \((3\sqrt5 + 5 \sqrt 2) \)

\[ \begin{align} (2\sqrt 5 - 4\sqrt 2)  +(3\sqrt5 + 5 \sqrt 2) &=2\sqrt 5 - 4\sqrt 2 +3\sqrt 5  + 5 \sqrt 2 \\ &=5\sqrt 2– 4 \sqrt 2 +2\sqrt 5 +3\sqrt 5   \\ &=\sqrt 2 +5\sqrt 5  \end{align} \]

Find the product of the surds \((\sqrt 2 + 5\sqrt 3)\) ,  \((\sqrt 3 – 2\sqrt 5) \)

\[ \begin{align} (\sqrt 2 + 5\sqrt 3) \times (\sqrt 3 – 2\sqrt 5)& =\sqrt 2(\sqrt 3 – 2\sqrt 5) +5\sqrt3(\sqrt 3 – 2\sqrt 5) \\ &=\sqrt 2 \times \sqrt 3 - \sqrt 2 \times  2\sqrt 5 +5\sqrt3 \times\sqrt 3 -5\sqrt3\times 2\sqrt 5 \\ &= \sqrt6 -2\sqrt10-5\sqrt9 -10\sqrt15 \\ &= -5\times3+\sqrt6 -2\sqrt10 -10\sqrt15 \\&=-15+\sqrt6 -2\sqrt10 -10\sqrt15 \end{align} \]

Find the square root  of \(\sqrt {5 + 2\sqrt6}\)

Hint: You may use the formula  \((a + b)^2 = a^2 + b^2 + 2ab\), and here we have \(a^2 + b^2 = 5; ab = 6 \)

Interactive Questions on Surds

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Let's Summarize

The mini-lesson targeted the fascinating concept of surds. The math journey around surds starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

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FAQs on Surds

1. what are surds in math.

Surds definition in math refers to the numbers that do not have answers to their roots. A few examples of surds as \(\sqrt5,~ \sqrt[3]7,~ 2 +\sqrt3,~ \sqrt6 + 2\sqrt3 \).

2. How to rationalize surds?

To rationalize a surd we need to multiply the surd with its conjugate surd. To rationalize \(\sqrt5 \) we need to multiply it with \(\sqrt5 \).  \[\sqrt5 \times \sqrt5 = 5 \]

3. How to solve surds?

To solve an expression in surds form we need to take prime factors of the number within the surd. Further, it is simplified by taking the possible prime factors outside the root symbol. \[\sqrt18 = \sqrt(3 \times 3 \times 2) = 3\sqrt2 \]

4. What is a surd example?

Some of the surds examples are \(\sqrt11,~ 5\sqrt3,~ 17 + \sqrt3,~ \sqrt5 + \sqrt10\)

5. Can surds have negative numbers?

A surds form does not have negative numbers. Surds can only have positive numbers, decimals, and fractions.

6. Can a surd be a fraction?

A surds form can be a fraction also. A few examples of surd fractions are as follows. \[\sqrt{\frac{3}{2}}, \frac{1}{\sqrt5} \]

7. What are the types of surds?

There are three types of surds. Simple surds, mixed surds, and compound surds.

Simple surd is of the form \(\sqrt x \). Some of the examples are \(\sqrt 5, ~\sqrt 0.8 ,~ \sqrt 17 \)

Mixed surd are of the form \(x\sqrt y \). A few examples are \(5\sqrt3,~2\sqrt7,~ 3\sqrt2 \)

Compound surd is a sum of two or more surds. Examples include \(5 + \sqrt2,~4\sqrt3 + 3\sqrt5,~ 5 + \sqrt3 + \sqrt11 \)

8.  What are conjugate surds?

Conjugate surds are the surds which, on multiplying with the given surd, convert it into a non-surd.

Conjugate surd of \( \sqrt5 \) is \(\sqrt 5 \)

Conjugate surd of \((2 + \sqrt 3 ) \) is \((2 - \sqrt 3) \)

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Surds at a glance

Surds are values written as square roots that cannot be further simplified. Surds are used to ensure very exact values are given in calculations. Surds are irrational numbers, if written as decimals, they would go on perpetually with no pattern. If written as fractions irrational numbers cannot have integer numerators or denominators.

We can multiply and divide surds. When multiplying surds, we use the rule “root a multiplied by root b = the root of a multiplied by b”. Similarly, when dividing surds we use the rule “root a divided by root b = the root of a divided by b”. 

We can sometimes simplify surds by taking out factors which are square numbers. We need to keep simplifying the surd until the number under the root sign has no factors which are square numbers.

To add and subtract surds the number under the root sign must be the same. We can only add or subtract two surds with different numbers under the roots if we can simplify one or both of them so that they have the same number under the root. 

Sometimes we might be asked to rationalise the denominator of a fraction. This means finding an equivalent fraction which does not have a surd in the denominator.

Looking forward, students can then progress to additional number worksheets , for example a fractions worksheet or a rounding worksheet .

For more teaching and learning support on Number our GCSE maths lessons provide step by step support for all GCSE maths concepts.

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GCSE Surds Questions and Answers

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Surds are numbers that have been left in square root form and are utilized when precise calculations are necessary. They are figures that, if expressed in decimal form, would go on forever.

1.1 Rational Numbers

A number that can be written as an integer (whole number) or a simple fraction is called a rational number. Rational numbers can be terminating decimals or recurring decimals. 

For example: 2, 100, -3, 2⁄11 .

1.2 Irrational Numbers

Irrational numbers in decimal form are infinite, with no recurring or repeating pattern. E.g. π is an example of an irrational number,

When a root (square root, cube root or higher) gives an irrational number, it is called a surd. 

For example: √9 = 3, which is an integer. The square root of 9 is not a surd. 

√5 = 2.23606, which is an infinitely long decimal with no recurring or repeating pattern, i.e. an irrational number. The square root of 5 is a surd.

1.3 What do you mean by surds?

Surds are irrational numbers that are left as square roots. An irrational number cannot be expressed as a fraction, and it would be infinitely long in decimal form with no recurring  pattern.

Surds can be a square root, cube root, or other root and are used when detailed accuracy is required in a calculation. 

The examples of surds are √ 2 , √ 3 , √ 5 , etc., as these values cannot be further simplified. If we further simply them, we get decimal values, such as:

                       √2 = 1.4142135…

                       √3 = 1.7320508…

                       √5 = 2.2360679…

1.4 How to simplify surds?

In order to simplify a surd, follow these steps: 

• Find a square number that is a factor of the number under the root. 
• Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number. 
• Repeat if the number under the root still has square factors.

Example: Simplify √ 24

Example: Simplify 3 x 4√54

Solution: 

1.5 Adding and subtracting surds

In order to add/subtract surds, we follow these steps:

• Check whether the terms are 'like surds'.
• If they aren't like surds, simplify each surd as far as possible.
• Combine the like surd terms by adding/subtracting.

Example - Simplify : √ 45 - 2√ 5

1.6 Multiplying and Dividing surds

Multiplying surds with the same number inside the square root 

We know that: (√3)² = √3 x √3 = √9 = 3

In order to multiply/divide surds, we follow these steps:

• Simplify the surds if possible.
• Use surd laws to fully simplify the numerator and denominator of the fraction. 
• Divide the numerator by the denominator.

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Exam Questions on Surds

These lessons, with videos, examples and step-by-step solutions, help GCSE Maths students learn about surds by working through some examination questions.

Related Pages More Sample Exam Questions On Surds More Lessons for GCSE Maths Math Worksheets

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How to do Surds? GCSE A/A* Maths revision Higher level exam questions (simplifying, rationalising)

GCSE Maths Surds A / A* Tutorial explaining surds for GCSE, including past exam paper question examples. This video can also be used for understanding the basics of surds for A-level Maths.

Simplifying Surds

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Problems on Surds

We will solve different types of problems on surds.

1. State whether the following are surds or not with reasons:

(i) √5 × √10

(ii) √8 × √6

(iii) √27 × √3

(iv) √16 × √4

(v) 5√8 × 2√6

(vi) √125 × √5

(vii) √100 × √2

(viii) 6√2 × 9√3

(ix) √120 × √45

(x) √15 × √6

(xi) ∛5 × ∛25

= \(\sqrt{5\cdot 10}\)

= \(\sqrt{5\cdot 5\cdot 2}\)

= 5√2, which is an irrational number.  Hence, it is a surd.

= \(\sqrt{8\cdot 6}\)

= \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 3}\)

= 4√3, which is an irrational number.  Hence, it is a surd.

= \(\sqrt{27\cdot 3}\)

= \(\sqrt{3\cdot 3\cdot 3\cdot 3}\)

= 9, which is a rational number.  Hence, it is not a surd.

= \(\sqrt{16\cdot 4}\)

= \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2}\)

= 2 × 2 × 2

= 8, which is a rational number.  Hence, it is not a surd.

= 5 × 2 \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 3}\)

= 10 × 2 × 2 × √3

= 40√3, which is an irrational number.  Hence, it is a surd.

= \(\sqrt{125\cdot 5}\)

= \(\sqrt{5\cdot 5\cdot 5\cdot 5}\)

= 25, which is a rational number.  Hence, it is not a surd.

= \(\sqrt{100\cdot 2}\)

= \(\sqrt{2\cdot 2\cdot 5\cdot 5\cdot 2}\)

= 2 × 5 × √2

= 10√2, which is an irrational number.  Hence, it is a surd.

= 6 × 9 \(\sqrt{2\cdot 3}\)

= 54√6, which is an irrational number.  Hence, it is a surd.

= \(\sqrt{120\cdot 45}\)

= \(\sqrt{2\cdot 2\cdot 2\cdot 3\cdot 5\cdot 3\cdot 3\cdot 5}\)

= 2 × 3 × 5 × √6

= 30√6, which is an irrational number.  Hence, it is a surd.

= \(\sqrt{15\cdot 6}\)

= \(\sqrt{3\cdot 5\cdot 2\cdot 3}\)

= 3√10, which is an irrational number.  Hence, it is a surd.

= \(\sqrt[3]{5 × 25}\)

= \(\sqrt[3]{5 × 5 × 5}\)

= 5, which is a rational number.  Hence, it is not a surd.

2.  Rationalize the denominator of the surd \(\frac{√5}{3√3}\).

\(\frac{√5}{3√3}\)

= \(\frac{√5}{3√3}\) × \(\frac{√3}{√3}\)

= \(\frac{\sqrt{5 \times 3}}{3 \times \sqrt{3 \times 3}}\)

= \(\frac{√15}{3 × 3}\)

= \(\frac{1}{9}\)√15

3.  Rationalize the denominator of the surd \(\frac{2}{√7 - √3}\)

\(\frac{2}{√7 - √3}\)

= \(\frac{2 × (√7 + √3)}{(√7 - √3) × (√7 + √3)}\)

= \(\frac{2 (√7 + √3)}{7 - 3}\)

= \(\frac{2 (√7 + √3)}{4}\)

= \(\frac{(√7 + √3)}{2}\)

4.  Express the surd \(\frac{√3}{5√2}\) in the simplest form.

\(\frac{√3}{5√2}\)

= \(\frac{√3}{5√2}\) × \(\frac{√2}{√2}\)

= \(\frac{\sqrt{3 \times 2}}{5 \times \sqrt{2 \times 2}}\)

= \(\frac{√6}{5 × 2}\)

= \(\frac{1}{10}\)√6, is the required simplest form of the given surd.

5.  Expand (2√2 - √6)(2√2 + √6), expressing the result in the simplest form of surd:

(2√2 - √6)(2√2 + √6)

= (2√2)\(^{2}\) - (√6)\(^{2}\), [Since, (x + y)(x - y) = x\(^{2}\) - y\(^{2}\)]

6. Fill in the blanks:

(i) Surds having the same irrational factors are called ____________ surds.

(ii) √50 is a surd of order ____________.

(iii) \(\sqrt[9]{19}\) × \(\sqrt[5]{10^{0}}\) = ____________.

(iv) 6√5 is a ____________ surd.

(v) √18 is a ____________ surd.

(vi) 2√7 + 3√7 = ____________.

(vii) The order of the surd 3∜5 is a ____________.

(viii) ∛4 × ∛2 in the simplest form is = ____________.

(i) similar.

(iii) \(\sqrt[9]{19}\), [Since, we know, 10\(^{0}\) = 1]

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Surds - Problem Solving

Subject: Mathematics

Age range: 14 - 18

Resource type: Lesson (complete)

Last updated

24 March 2019

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Lesson looks at using surds with within questions such as finding areas and perimeters of shapes. Finding missing sides in right-angled triangles. Finding surface areas and volumes. Worked examples. All answers included.

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Resourceaholic

Ideas and resources for teaching secondary school mathematics

• Blog Archive

19 May 2014

• Pupils list the first twelve square numbers in their books for reference throughout the lesson. They need to readily recognise square numbers in order to simplify surds so the more they practise listing them, the better.
• When manipulating expressions containing surds (eg expanding brackets) I make comparisons to what they already know about algebra. For example √2 and √3 can be thought of in the same way as x and y (ie not ‘like terms’) whereas 2√3 and 5√3 can be thought of in the same way as 2x and 5x (ie we can add them to get 7√3).
• Ask pupils to find a way of drawing a line with a length of exactly √5 units (the hypotenuse of a right angled triangle with sides 2cm and 1cm) 
• Ask pupils to divide the length of an A4 piece of paper by its width. Repeat for A3 and A5. What do they notice? (The answer is always √2)
• Is it rational?
• Two interactive tasks from Mathspad
• Pairs activity 
• Introducing surds
• True or false activity
• Surds -Applying and problem solving
• Multiplication squares
• Mr Barton’s activity
• Standards Unit N11
• Surds Connect 4
• Surds resources - leannegadsby on TES
• Manipulating Radicals
• And here’s some resources recommended by the TES .

5 comments:

What a fabulous summary of several lesson plans!

I wrote more about surds here: http://www.resourceaholic.com/2018/07/oldsurds.html

Thanks for these. This year I've started teaching simplifying surds using a prime factor tree. It works perfectly and removes the need for a list of square factors.

The factor tree method is one that I was shown by one of my students this year. This comes after teaching this topic for the past 5 years, and learning it back in school. Just goes to show how we're always learning...

Agree! It's great to be shown new methods.