Learning Materials
Features
Discover
Surds are expressions that contain a square root, cube root or other roots. They are roots of numbers that produce an irrational number as a result, with infinite decimals. Therefore, they are left in their root form to represent them more exactly. For example, \(\sqrt2, \sqrt 3, \sqrt5, \sqrt 6, \sqrt 7, 2\sqrt 2\).
Millions of flashcards designed to help you ace your studies
Sign up for freeWhat is the correct simplification of \(\sqrt{20}\)?
Which of these is equivalent to \(7\sqrt{2}\)?
How would you rationalize the fraction \[ \frac{4}{\sqrt{2}}?\]
How can you simplify \[ \frac{ \sqrt{5} + 3}{\sqrt{5} - 2}?\]
Can you add \(\sqrt{2}\) and \(\sqrt{3}\)?
Can you add \(\sqrt{2}\) and \(\sqrt{50}\)?
What is the correct simplification of \(\sqrt{20}\)?
Which of these is equivalent to \(7\sqrt{2}\)?
How would you rationalize the fraction \[ \frac{4}{\sqrt{2}}?\]
How can you simplify \[ \frac{ \sqrt{5} + 3}{\sqrt{5} - 2}?\]
Can you add \(\sqrt{2}\) and \(\sqrt{3}\)?
Can you add \(\sqrt{2}\) and \(\sqrt{50}\)?
Achieve better grades quicker with Premium
Geld-zurück-Garantie, wenn du durch die Prüfung fällst
Did you know that StudySmarter supports you beyond learning?
Review generated flashcards
Start learning or create your own AI flashcards
Team Surds Teachers
Remember that an irrational number is a type of number that cannot be represented as a fraction.
To simplify surds, you need to remember the square roots of perfect squares. If the number inside the root of a surd has a square number as a factor, then it can be simplified.
\(\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt 4 \cdot \sqrt 3 = 2 \cdot \sqrt 3 = 2\sqrt3\)
The steps to follow to simplify surds are:
Write the number inside the root as the multiplication of its factors. One of the factors should be a square number.
\(\sqrt{12} = \sqrt{4 \cdot 3}\)
Split the factors into separate roots
\(\sqrt{4 \cdot 3} = \sqrt 4 \cdot \sqrt 3\)
Simplify the terms
\(\sqrt 4 \cdot \sqrt 3 = 2 \cdot \sqrt 3\)
Take out the multiplication symbol
\(2 \cdot \sqrt 3 = 2\sqrt3\)
When working with surds, you need to remember the following rules:
Multiplying surds: As long as the index of the roots is the same, you can multiply surds with different numbers inside the root by simply combining them into one root and multiplying the numbers inside the root. Likewise, you can split a root into separate roots using factors.
\(\sqrt a \cdot \sqrt b = \sqrt{a \cdot b}\)
\(\sqrt2 \cdot \sqrt 5 = \sqrt{2 \cdot 5} = \sqrt{10}\)
Dividing surds: Similarly, as long as the index of the roots is the same, you can divide surds with different numbers inside the root by combining them into one root and dividing the numbers inside the root.
\(\frac{\sqrt a}{\sqrt b} = \sqrt{\frac{a}{b}} \)
\(\frac{\sqrt {10}}{\sqrt 2} = \sqrt{\frac{10}{2}} = \sqrt5\)
Multiplying a square root by itself: If you multiply the square root of a number by itself, you should obtain the original value.
\(\sqrt{a} \cdot \sqrt{a} = (\sqrt{a})^2 = a\)
\(\sqrt{3} \cdot \sqrt{3} = (\sqrt{3})^2 = 3\)
Multiplying a number by a surd: When multiplying a number by a surd, the order of the factors does not matter, and the result should be the number followed by the surd.
\(a \cdot \sqrt{b} = \sqrt{b} \cdot a = a\sqrt{b}\)
\(3\cdot \sqrt{2} = \sqrt{2} \cdot 3 = 3\sqrt{2}\)
Adding or subtracting surds: To add or subtract surds, the number inside the roots must be the same. You add or subtract the numbers outside the root.
\(a\sqrt{x} + b\sqrt{x} = (a+b)\sqrt{x}\)
\(a\sqrt{x} - b\sqrt{x} = (a-b)\sqrt{x}\)
\(5\sqrt{3} + 2\sqrt{3} = (5+2)\sqrt{3} = 7\sqrt{3}\)
\(5\sqrt{3} - 2\sqrt{3} = (5-2)\sqrt{3} = 3\sqrt{3}\)
To add or subtract surds, you might need to simplify them first to find like terms.
You cannot add \(\sqrt{2} + \sqrt{8}\), but you can simplify \(\sqrt{8}\) first,
\(\sqrt 8 = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt2\)Then you can solve \(\sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}\)
Multiplying brackets containing surds: To multiply brackets containing surds, each term in the first bracket must be multiplied by each term in the second bracket. Then you can combine like terms.
The purpose of rationalizing the denominator of fractions containing surds is to eliminate the surd from the denominator. The strategy to do this is to multiply the numerator and denominator by the surd.
If the denominator contains only a surd:
Rationalize the denominator in the following expression: \(\frac{5}{\sqrt3}\)
Using the rules: \(a \cdot \sqrt{b} = \sqrt{b} \cdot a = a\sqrt b\) and \(\sqrt{a} \cdot \sqrt{a} = (\sqrt{a})^2 = a\)
\(\frac{5}{\sqrt3} = \frac{5}{\sqrt3} \cdot \frac{\sqrt3}{\sqrt3} = \frac{5\sqrt3}{(\sqrt3)^2} = \frac{5\sqrt3}{3}\)
If the denominator contains a surd and a rational number: In this case, you need to multiply the numerator and denominator by the expression in the denominator, but with the sign in the middle changed, ie if it is (+) change it to ( -) and vice versa. This expression is called the conjugate.
Rationalize the denominator in the following expression: \(\frac{(\sqrt6 + 3)}{(\sqrt6 -2)}\)
The conjugate of \((\sqrt6 - 2)\) is\((\sqrt6 + 2)\)
Multiplying the brackets and combining like terms, you can see that the surds in the denominator cancel each other.
\(\frac{(\sqrt6 + 3)}{(\sqrt6 -2)} \cdot \frac{(\sqrt6 + 2)}{(\sqrt6 +2)} = \frac{6 +2\sqrt6+3\sqrt6 +6}{6 +\cancel{2\sqrt6 -2\sqrt6}-4} = \frac{12 +5\sqrt6}{2}\)
Surds are expressions that contain a square root, cube root or other roots, which produce an irrational number as a result, with infinite decimals. They are left in their root form to represent them more precisely.
To multiply and divide surds with different numbers inside the root, the index of the roots must be the same.
To add or subtract surds, the number inside the roots must be the same.
To add or subtract surds, they might need to be simplified first.
If the number inside the root of a surd has a square number as a factor, then it can be simplified.
The purpose of rationalizing the denominator of fractions containing surds is to eliminate the surd from the denominator.
Sign up for free to gain access to all our flashcards.
What is a surd?
Surds are expressions that contain a square root, cube root or other roots, which produce an irrational number as a result, with infinite decimals. They are left in their root form to represent them exactly.
What is an example of a surd?
√2, √3, and 2√2 are examples of surds.
What are the rules of surds?
a√x + b√x = (a + b)√x
a√x - b√x = (a - b)√x
How do you add surds?
To add or subtract surds, the number inside the roots must be the same.
a√x + b√x = (a + b)√x
How do you simplify surds?
The steps to simplify surds are:
At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified.
Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy.
Get to know Lily
Gabriel Freitas is an AI Engineer with a solid experience in software development, machine learning algorithms, and generative AI, including large language models’ (LLMs) applications. Graduated in Electrical Engineering at the University of São Paulo, he is currently pursuing an MSc in Computer Engineering at the University of Campinas, specializing in machine learning topics. Gabriel has a strong background in software engineering and has worked on projects involving computer vision, embedded AI, and LLM applications.
Get to know Gabriel
StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.
Learn moreSave explanations to your personalised space and access them anytime, anywhere!
Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.
Already have an account? Log in
Sign up to highlight and take notes. It’s 100% free.
Explanations 
Flashcards 
StudySmarter AI 
Notes 
Study Plans 
Study Sets 
Exams 
Find a job 
Student Deals 
Magazine 
Mobile App 
