Surds is one of the more intimidating topics on the GCSE Maths curriculum. Put simply, a surd represents irrational numbers that can't be simplified into an exact decimal/fraction. An example is the square root of 3. This guide will explain what surds are, how to simplify them and how we can write operations with them. We'll give you all the knowledge needed to tackle questions that include surds in your exams.
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Understanding Surds
Surds are irrational numbers that can't be simplified to remove the root sign. They stay in root form because the decimal representation is infinite and non-repeating, meaning it's not possible to express as a precise fraction or terminating decimal. An example is the square root of 5. Surds help us to express exact values without using long decimal approximations. They also simplify complex expressions, especially in algebraic and geometric calculations.
Surds are normally the root of non-perfect squares or higher-order roots that can't be simplified into rational numbers. For example:
- The square root of 4 is 2, a rational number because it can be expressed as a simple fraction (2/1).
- However, the square root of 2, begins with a value of 1.414213... It's a non-repeating, non-terminating decimal. This makes it irrational and a surd.
Surds appear frequently in maths, including geometry (such as calculating the length of the sides of a right-angled triangle using Pythagoras' theorem) and algebra.
How to recognise a Surd
Surds are normally identified by their root symbol. They are often part of expressions that can't be simplified into rational numbers. These are common surds:
- The square root of 2
- The square root of 3
- The square root of 5
- The square root of 7
These are in surd form to keep their accuracy because they can't be simplified into an exact fraction.
Simplifying surds
Simplifying surds down to their simplest form helps make them easier to understand and work with. A surd can be simplified until there are no perfect square factors left under the root sign:
- Find the perfect square factors: Look for factors of the number under the root that are perfect squares.
- Rewrite the surd: Express the surd as a product of the square root of the perfect square and another number.
- Simplify: Evaluate the square root of the perfect square and simplify the expression.
Let's say we need to simplify the square root of 50:
- First, look for the perfect square factors of 50. The largest perfect square is 25: 25 x 2 = 50.
- Rewrite the square root of 50 as the square root of(25 × 2).
- The square root of 25 = 5 so we can simplify it to 5 times the square root of 2.
Combining Like Surds
When combining surds, we need to follow the rule that you can only add/subtract like term. Like surds have the same number under the root:
- 3 times the square root of 2 and 4 times the square root of 2 can be combined into 7 times the square root of 2.
- 3 times the square root of the square root of 2 and 5 times the square root of 4 can't be combined because they are unlike surds.
Combining surds is similar to algebra: the coefficients (numbers in front of the surds) are added/subtracted and the surd part remains the same.
Example: Combine 2 times the square root of 3 + 3 times the square root of 3.
- Both the surds are the square root of 3 so we can add the coefficients to 5 times the square root of 3.
1
Which is the simplified form of the square root of 72?
Simplifying Surds
You are expected to simplify surds in GCSE Maths so that no perfect square factors are left under the root sign. The only exception is the value 1. These are the steps to simplify a surd:
- Find the perfect square factors - Look for the largest perfect square that divides the number under the root. Perfect squares are numbers like 1, 4, 9, 16, 25, 36 etc.
- Rewrite the Surd - Express the surd as a product of the square root of the perfect square and another number. This requires you to break the number down into its factor components.
- Simplify the expression - Take the square root of the perfect square and keep lowering the number under the root until it can't be reduced by any perfect square factors.
Simplifying surds is used to solve quadratic equations as well as working with trigonometric identities and geometry. For example, surds often represent the lengths of the sides of triangles when using the Pythagorean theorem.
Simplifying surds examples
Simplify the square root of 72:
- Find the largest perfect square factor of 72, which is 36 (36 x 2 = 72).
- Change the square root of 72 to the square root of (36 × 2).
- The square root of 36 is 6 so we can simplify this to 6 times the square root of 2.
Simplify the square root of 128:
- The largest perfect square factor is 64 (64 x 2 = 128).
- The square root of 128 is simplified to the square root of (64 × 2).
- The square root of 64 is 8 (8 x 8 = 64) so we can simplify further to 8 times the square root of 2.
Simplify the square root of 75:
- 25 is the largest perfect square factor of 75.
- Rewrite the square root of 75 to the square root of (25 × 3).
- Simplify this further to 5 times the square root of 3 as the square root of 25 is 5 (5 x 5 = 25).
Combining and simplifying
Some expressions will have multiple surds. You need to simplify each surd one at a time first and then combine them.
For example, simplify 3 times the square root of 50 + 2 times the square root of 18:
- Simplify each surd:
- The square root of 50 = the square root of (25 × 2) = 5 times the square root of 2
- The square root of 18 = the square root of (9 × 2) = 3 times the square root of 2
- Add the simplified surds into the expression:
- 3 times the square root of 50 = 3 × 5 times the square root of 2 = 15 times the square root of 2
- 2 times the square root of 18 = 2 × 3 times the square root of 2 = 6 times the square root of 2
- Combine the like surds:
- 15 times the square root of 2 + 6 times the square root of 2 = 21 times the square root of 2
2
Which is the simplest form of the square root of 98?
Operations with Surds
Now we have a good understanding of surds, we can start performing operations with them. This includes adding surds, subtracting, multiplying and dividing them. This is important for solving complex algebraic problems.
Adding and subtracting Surds
Adding and subtracting surds is similar to combining like terms in algebra. You can only add or subtract surds that have the same root value. These surds are called like surds.
Adding Like Surds examples:
- Simplify 3 times the square root of 5 + 2 times the square root of 5 | Both terms have the same surd. This means we can add the coefficients: 3 times the square root of 5 + 2 times the square root of 5 = 5 times the square root of 5.
Subtracting Like Surds example:
- Simplify 7 times the square root of 3 - 4 times the square root of 3 | Both of these terms have the same surd so we can subtract the coefficients: 7 times the square root of 3 - 4 times the square root of 3 = 3 times the square root of 3.
When the surds are not like surds, they can't be added or subtracted. These are called unlike surds.
Adding Unlike Surds example:
- Simplify 2 times the square root of 3 + 3 times the square root of 2 | Here the surds are different (the square root of 3 and the square root of 2). This means they can't be combined and 2 times the square root of 3 + 3 times the square root of 2 remains as it is
Multiplying and dividing Surds
When we divide and multiply surds, we need to use the distributive property and the rules of exponents. These operations help simplify expressions, including complex algebraic expressions. They are also used to solve equations.
You can multiply the coefficients (the numbers outside the surds) and the numbers inside the surds separately.
Multiplying simple Surds example:
- Simplify (2 times the square root of 3) × (4 times the square root of 5) | Multiply the coefficients (2 x 4 = 8) and the surds (the square root of 3 x the square root of 5 = the square root of 15) to get 8 times the square root of 15.
Multiplying Like Surds example:
- Simplify (3 times the square root of 2) × (2 times the square root of 2) | Multiply the coefficients (3 x 2 = 6) and the surds (the square root of 2 x the square root of 2 = the square root of 4): (6)(the square root of 4) = 6 × 2 = 12. Here, the result is a rational number.
Dividing Surds
We need to rationalise the denominator when dividing surds. This is important to ensure no surds are left in the denominator.
Simple division of Surds example:
- Simplify the square root of 12 / the square root of 3 | Divide the numbers under the surds: the square root of 12 / the square root of 3 = the square root of 12 divided by 3 = the square root of 4 = 2.
Rationalising the denominator example:
- Simplify 5 / the square root of 2 | Multiply the numerator and the denominator by the square root of 2: 5 / the square root of 2 × the square root of 2 / the square root of 2 = 5 times the square root of 2 / 2.
Combining operations with Surds
Sometimes, you will need to combine multiple operations with surds. When this happens, you need to follow the order of operations called BIDMAS for accurate results. This is short for Brackets, Indices, Division and Multiplication, Addition and Subtraction.
Combined Operations example:
- Simplify 2 times the square root of 3 × (4 times the square root of 3 - the square root of 5) | Use the distributive property: 2 times the square root of 3 × 4 times the square root of 3 - 2 times the square root of 3 × the square root of 5. Simplify each term: 8 × 3 - 2 times the square root of 15 = 24 - 2 the square root of 15.
3
Simplify (3 times the square root of 2) × (2 times the square root of 5).
Rationalising the denominator
You need to understand how to express fractions without surds in the denominator. This is called rationalising the denominator, which simplifies the expression to allow for future calculations. Rationalising the denominator helps to simplify complex fractions in algebra and calculus. It is also used in geometry to solve length and distance problems written in surd form.
Steps to rationalise the denominator
- Find the Surd in the denominator: Check if the surd is a simple root or part of a complex binomial.
- Multiply by the conjugate: If you have binomials (expressions with two terms), use the conjugate of the denominator. The conjugate of a+ba + ba+b is a times the square root of ba - ba times the square root of b and vice versa.
- Multiply numerator and denominator: Use the same factor to multiply the numerator and the denominator. This makes sure the value of the fraction stays the same.
- Simplify the expression: Simplify the numerator and the denominator as much as possible after multiplying.
For example, rationalise the simple surd 5 / the square root of 3:
- The surd in the denominator is the square root of 3.
- Multiply the numerator and the denominator by the square root of 3 to remove the surd from the denominator: 5/ the square root of 3 x the square root of 3 / the square root of 3 = 5 times the square root of 3 / the square root of 3.
For example, simplify the binomial denominator 3 / the square root of 2 + 1:
- The surd in the binomial denominator is the square root of 2+1.
- Multiply the numerator and the denominator by the conjugate of the denominator, which is the square root of 2 - 1: 3 / the square root of 2 + 1 x the square root of 2 - 1 / the square root of 2 - 1 = 3(the square root of 2 - 1) / (the square root of 2 + 1)(the square root of 2 - 1).
- Simplify the denominator using the difference of squares formula (a+b)(a times the square root of b)= a2 times the square root of b2: (the square root of 2+1)(the square root of 2 - 1) = (the square root of 2)2 - (1)2 = 2 - 1 = 1.
- Simplify the fraction: 3 (the square root of 2 - 1) / 1 = 3 times the square root of 2 - 3.
Example 3: Rationalising with complex binomials
Simplify 4 / 1 - the square root of 5:
- The surd in the denominator is 1 - the square root of 5.
- Multiply the numerator and the denominator by the conjugate of the denominator, 1 + the square root of 5: 4 / 1 - the square root of 5 x 1 + the square root of 5 / 1 + the square root of 5 = 4(1 + the square root of 5) / (1 - the square root of 5)(1 + the square root of 5).
- Simplify the denominator using the difference of squares formula: (1 - the square root of 5)(1 + the square root of 5) = 1 - 5 = -4.
- Simplify the fraction: 4(1 + the square root of 5) / the square root of 4 = -1 - the square root of 5.
4
Simplify 7 / 2 + the square root of 3?.
Applying Surds in GCSE exam questions
We have listed common exam questions for GCSE Maths that use surds. These questions test your ability to simplify surds, use them with operations and rationalising the denominator.
Common exam questions with Surds
- Simplifying Surd expressions - These questions will provide a simple surd expression to simplify as much as possible. You will need to find perfect square factors and express the surd in a simplified form.
- Operations with Surds - Questions could include all operations: adding, subtracting, multiplying, or dividing surds. Use the rules discussed above to combine like surds and simplify the expressions.
- Rationalising the denominator - You will be required to remove surds from the denominator of a fraction. This normally involves multiplying by the conjugate and simplifying expressions.
- Solving equations with Surds - There could be algebraic equations with surds, such as isolating the surd on one side and squaring both sides of the equations to remove the surd.
Question: Simplify the square root of 48:
- Find the largest perfect square factor, which is 16 (16 × 3 = 48).
- Simplify the square root of 48 to the square root of 16 x the square root of 3.
- Simplify further using 4, which is the square root of 16. The square root of 48 = 4 times the square root of 3.
Question: Simplify 2 times the square root of 5 x the square root of 20:
- Simplify the square root of 20. The square root of 20 = the square root of 4 x 5 = 2 times the square root of 5.
- Rewrite the expression. 2 times the square root of 5 x 2 times the square root of 5.
- Multiply the coefficients and the surds. (2×2)(the square root of 5 × the square root of 5) = 4 × 5 = 20.
Question: Simplify 3 / the square root of 2+1.
- Multiply the numerator and the denominator by the conjugate of the denominator (the square root of 2-1). 3(the square root of 2-1) / (the square root of 2)2 - 12 = 3(the square root of 2-1) / 2 - 1.
- Add the difference of squares to the denominator. 3(the square root of 2 - 1) / (the square root of 2)2 - 12 x 3(the square root of 2 - 1) / 2 - 1.
- Simplify the denominator to 1. 3(the square root of 2 - 1) = 3 times the square root of2 - 3.
How to tackle Surd questions in exams
- Simplify Surds early: Start any question by simplifying all surds to simplify the expression. Then you can approach the calculations.
- Check for Like Surds: Make sure the surds are like surds before you try to add or subtract. If they are unlike surds, you can't use these operations.
- Rationalise: Always rationalise fractions with surds in the denominator. Use the conjugate for binomials to simplify complex expressions.
- Practice algebraic techniques: Surds often appear in algebra so you need to be comfortable squaring both sides of an equation and using the difference of squares.
- Stay organised: Be methodical in your workings. Write each step clearly to avoid mistakes and see the process.
Conclusion
Surds can seem complicated a first. With practice and a methodical approach, you can start to get the hang of them for your GCSE exam. Here are some final tips to help you master surds:
- Use examples: Study a variety of examples of how to simplify surds and use them in operations. Start with the simple and work your way to more complex examples to give you a better understanding of the topic.
- Test yourself: There is a wealth of online resources you can lean on. These include step-by-step guides and tests at BBC Bitesize and Khan Academy.
- Seek help if needed: If you've put your all into learning this topic and continue to struggle, you may want to look for personalised support. Tutors can give you targeted advice to help you understand surds and similar topics in GCSE Maths.
Surds are a useful tool that allows precise solutions to a variety of problems. Getting the hang of this concept will help if you are looking to pursue advanced mathematics at A Level or beyond.
If you need expert guidance, view TeachTutti's list of GCSE Maths tutors. Our tutors can explain surds and other areas of GCSE Maths to help you prepare for your examinations.
Glossary
- Surd - An irrational number that can't be simplified to remove the root symbol. Surds are left in root form (e.g.the square root of
2) because their decimal form is infinite and non-repeating. - Rational number - A number that can be expressed as a fraction of two integers e.g. 1/3. They include integers, fractions and terminating or repeating decimals.
- Irrational number - A number that can't be written as a simple fraction and has non-terminating, non-repeating decimal expansions. Examples include pi and the square root of 2.
- Perfect square - An integer that is the square of another integer. For example, 16 is a perfect square because it is 4².
- Simplifying surds - Breaking a surd into its simplest form by factoring out perfect squares from under the root sign e.g.the square root of
50 simplifies to 5 times the square root of 2. - Like surds - Surds that have the same number under the root. The square root of 3 and 2 the square root of 3 are like surds because both involve the square root of 3. Like surds can be added or subtracted by combining their coefficients.
- Coefficient - A constant factor that multiplies a variable/surd in a term.
- Binomial - An algebraic expression with two terms, such as a+b. Binomials are often used with surds, especially when rationalising denominators.
- Expression - A combination of numbers, variables and operations (including addition, subtraction, multiplication and division) that represent a value. For example, 2 times the square root of 3 + 4.
- Square root - A value that returns the original number when multiplied by itself e.g. the square root of 9 is 3 (3 x 3 = 9). The square root of a non-perfect square is often left in surd form (e.g. 2 as the square root of 2).
- Exponent - A number telling how many times to multiply the base number by itself. For example, the exponent in 23 is 3, which means 2×2×2.
