Basic 9 · Mathematics · SURDS
Surds and Their Rules
In this topic you will identify simple and compound surds, simplify them and explain the rules of surds.
By the end of these notes you should be able to:
- Identify simple and compound surds
- Simplify surds using perfect squares
- Explain and apply the rules of surds
What you already know
- Square roots
- Perfect squares
Copy everything that has a blue line on its left into your notebook. The other parts are for reading.
Lesson 1
- Strand
- Number
- Sub-strand
- SURDS
- Indicator
- B9.1.2.4.1 – Identify simple and compound surds
Introduction
√9 is exactly 3. √2 has a decimal that goes on forever without repeating. That difference is what makes it a surd.
1. What Is a Surd?
Surd: A root of a number that cannot be simplified to remove the root sign, and so is irrational.
| Number | Value | Surd? |
|---|---|---|
| √4 | 2 | no |
| √9 | 3 | no |
| √2 | 1.414213... | yes |
| √5 | 2.236067... | yes |
| √(1/4) | ½ | no |
| ∛8 | 2 | no |
| ∛7 | 1.912931... | yes |
A surd has an endless, non-repeating decimal. If we write 1.414, we have only an approximation. Keeping √2 in surd form is exact. This is why mathematicians and engineers often leave answers in surd form.
2. Simple and Compound Surds
| Type | Meaning | Examples |
|---|---|---|
| simple | one term | √3, 5√2 |
| compound | two or more terms | 2 + √3, √5 − √2 |
| like | same number under the root | 2√3 and 5√3 |
| unlike | different numbers under the root | √2 and √3 |
In 5√2, the 5 is the coefficient, and it multiplies the surd. The number under the root sign is called the radicand. Like surds can be added and subtracted in the same way as like terms in algebra.
3. Where Surds Come From
Draw a square with sides of 1 unit. By Pythagoras' theorem, the diagonal d satisfies d² = 1² + 1² = 2, so d = √2. Here is a real length that cannot be written exactly as a fraction. Long ago, Greek mathematicians were surprised to discover such numbers. Surds appear often in geometry, in construction and in physics.
4. Testing for a Surd
To decide whether a square root is a surd, check whether the number under the root is a perfect square. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144 are perfect squares. So √49 is not a surd, but √50 is. A calculator may show 7.0710678, but those digits continue forever. Do not confuse surds with recurring decimals, which are rational.
New words
- surd
- An irrational root.
- radicand
- The number under the root sign.
- coefficient
- The number multiplying a surd.
- compound surd
- A surd with two or more terms.
Summary
- A surd is a root that cannot be simplified exactly.
- Surds may be simple or compound, like or unlike.
- Perfect squares are not surds.
Class activity: Surd Sort
- Write ten roots on cards.
- Sort them into surds and non-surds.
- Check with a calculator.
You need: Number cards, Square-grid paper, Calculator
Exercises
Answer in your exercise book first, then check your answers.
1. Which is a surd?
- A. √7
- B. √9
- C. √16
- D. √25
Show answer
Answer: √7Not a perfect square.2. Which are like surds?
- A. 2√3 and 5√3
- B. √2 and √3
- C. √5 and 5
- D. 2 and √2
Show answer
Answer: 2√3 and 5√3Same radicand.3. √36 is a surd.
- A. True
- B. False
Show answer
Answer: FalseIt equals 6.4. The number under a root sign is the ______.
Show answer
Answer: radicandTerm.5. Give an example of a compound surd.
Show answer
Answer: 3 + √2Two terms.6. Explain why √2 is a surd.
Show answer
Answer: 2 is not a perfect square, and its decimal is endless and non-repeating.Reason.
Lesson 2
- Strand
- Number
- Sub-strand
- SURDS
- Indicator
- B9.1.2.4.1 – Identify simple and compound surds
Introduction
√50 looks awkward, but inside it there is a perfect square waiting to be taken out.
1. The Method
- Find the largest perfect square factor of the number.
- Write the root as a product of two roots.
- Take the square root of the perfect square.
- Write the answer in simplest form.
| Surd | Working | Simplest form |
|---|---|---|
| √8 | √(4 × 2) | 2√2 |
| √12 | √(4 × 3) | 2√3 |
| √18 | √(9 × 2) | 3√2 |
| √45 | √(9 × 5) | 3√5 |
| √50 | √(25 × 2) | 5√2 |
| √72 | √(36 × 2) | 6√2 |
| √200 | √(100 × 2) | 10√2 |
2. Worked Examples
Simplify
Simplify √75.
- 75 = 25 × 3.
- √75 = √25 × √3.
Answer: 5√3
With a coefficient
Simplify 3√32.
- 32 = 16 × 2.
- 3 × √16 × √2 = 3 × 4 × √2.
Answer: 12√2
3. Why Simplify?
Simplifying makes surds easier to compare and combine. √8 and √18 look unrelated, but they are 2√2 and 3√2, which are like surds, and can be added. Simplest form also lets you see the size of a surd. In examinations, the final answer is usually expected in simplest form. Make sure that the number under the root has no perfect square factor left.
4. Using Prime Factors
If you cannot find the largest square, use prime factors. For √72: 72 = 2 × 2 × 2 × 3 × 3. Pairs of equal primes come out of the root as one number: a pair of 2s gives 2, and a pair of 3s gives 3. One 2 stays inside. So √72 = 2 × 3 × √2 = 6√2. This method always works, however large the number is. If you take out a smaller square first, you must continue: √72 = 2√18 = 2 × 3√2 = 6√2.
New words
- perfect square
- A number that is an integer squared.
- simplest form
- A surd with no square factor inside.
- prime factor
- A factor that is a prime number.
- factor
- A number that divides exactly.
Summary
- Take out the largest perfect square factor.
- Prime factors always work.
- Simplest form makes surds easier to use.
Class activity: Square Hunt
- Write the first twelve perfect squares.
- Use them to simplify ten surds.
- Check each other's work.
You need: Number cards, Square-grid paper, Calculator
Exercises
Answer in your exercise book first, then check your answers.
1. √12 simplifies to…
- A. 2√3
- B. 3√2
- C. 4√3
- D. 6
Show answer
Answer: 2√34 × 3.2. √98 =
- A. 7√2
- B. 2√7
- C. 9√2
- D. 49√2
Show answer
Answer: 7√249 × 2.3. Surds should be left in simplest form.
- A. True
- B. False
Show answer
Answer: TrueConvention.4. √20 = ______√5.
Show answer
Answer: 24 × 5.5. Simplify 2√27.
Show answer
Answer: 6√32 × 3√3.6. Simplify √128 using prime factors.
Show answer
Answer: 8√22⁷ gives three pairs.
Lesson 3
- Strand
- Number
- Sub-strand
- SURDS
- Indicator
- B9.1.2.4.2 – Explain the identities/rules of surds
Introduction
Like indices, surds follow simple rules. Learn them, and hard-looking expressions become easy.
1. The Rules
| Rule | Example |
|---|---|
| √a × √b = √(ab) | √2 × √8 = √16 = 4 |
| √a ÷ √b = √(a/b) | √18 ÷ √2 = √9 = 3 |
| (√a)² = a | (√5)² = 5 |
| m√c + n√c = (m + n)√c | 2√3 + 5√3 = 7√3 |
| m√c − n√c = (m − n)√c | 7√2 − 3√2 = 4√2 |
| a√b × c√d = ac√(bd) | 2√3 × 4√2 = 8√6 |
2. Important Warnings
You cannot add unlike surds. √2 + √3 is not √5. Check with a calculator: √2 + √3 ≈ 3.146, but √5 ≈ 2.236. Similarly, √(a + b) is not √a + √b, and √(9 + 16) = 5, not 7. These are the most common errors in surds. Only multiplication and division allow you to combine what is under the root sign.
3. Worked Examples
Adding
Simplify √12 + √27.
- √12 = 2√3 and √27 = 3√3.
- 2√3 + 3√3.
Answer: 5√3
Subtracting
Simplify √50 − √18.
- √50 = 5√2 and √18 = 3√2.
- 5√2 − 3√2.
Answer: 2√2
Multiplying
Simplify 3√2 × 2√5.
- 3 × 2 = 6.
- √2 × √5 = √10.
Answer: 6√10
4. Links to Indices
Surds and indices are closely linked. √a is the same as a^½, and this is why √a × √a = a^(½ + ½) = a¹ = a. It also explains why √(ab) = √a × √b, because (ab)^½ = a^½ × b^½. In senior high school, you will use these ideas to work with fractional indices. Mathematics is a connected web, and the rules are not separate facts. When you forget a rule, a simple test with numbers can help you to work it out again.
New words
- like surds
- Surds with the same radicand.
- unlike surds
- Surds with different radicands.
- rule
- A statement that is always true.
- index
- The power of a number.
Summary
- √a × √b = √(ab) and √a ÷ √b = √(a/b).
- Only like surds can be added or subtracted.
- Simplify surds first, then combine.
Class activity: True or False
- Write five statements about surds.
- Test them with a calculator.
- Explain which are false.
You need: Number cards, Square-grid paper, Calculator
Exercises
Answer in your exercise book first, then check your answers.
1. √3 × √3 =
- A. 3
- B. 9
- C. 6
- D. √6
Show answer
Answer: 3(√a)² = a.2. 3√5 + 2√5 =
- A. 5√5
- B. 5√10
- C. 6√5
- D. 6√25
Show answer
Answer: 5√5Like surds.3. √2 + √3 = √5.
- A. True
- B. False
Show answer
Answer: FalseUnlike surds.4. √2 × √18 = ______.
Show answer
Answer: 6√36.5. Simplify √8 + √32.
Show answer
Answer: 6√22√2 + 4√2.6. Simplify 2√3 × 5√12.
Show answer
Answer: 6010√36.
Homework
- Simplify ten surd expressions, showing your working.