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# Surds

Updated: Jun 22, 2021

Dear Secondary Math students, we will be going through Surds today. Surds will be covered in Secondary 3 A Math for both Express and Normal Academic stream. Math Lobby has summarised the important concepts of surds in this note. Without further ado, let's begin learning them!

In this note, you will learn:

1) What are surds?

2) Laws of surds

3) Mathematical operations of surds

### 1) What are surds?

By mathematical definition, surds are irrational (numbers that cannot be expressed in terms of a ratio of two integers/ in fractions) numbers that comprises of the square roots and cube roots. In this chapter, we will only be dealing with square roots. Examples of surds are: √2, √7, √139

However, square root of numbers like 4, 16, 25, etc. cannot be deemed as surds. Why is that so? Leave your answers in the comments down below before checking the answer to this question!

So, why are √4, √16 and √25 not counted as surds? It is because they are perfect squares! Perfect squares are numbers which are made from the product of a number multiplied by itself, hence the square root of a perfect square produces integers, which are rational numbers and cannot be surds as surds are irrationalnumbers! Examples of perfect squares are: √4 = 2, √16 = 4, √25 = 5.

### 2) Laws of surds

In an expression involving surds, having the knowledge on the laws of surds will definitely help us in simplifying the expressions. There are two general laws in this chapter of surds, which are:

Law 1: √ab = √a x √b,

Law 2: √(a/b) = √a / √b,

which a > 0, b > 0

After viewing these two laws, are you able to decipher the reason why they are as such? Leave your answers in the comments down below before viewing the answer!

The reason why the laws of surds are the way it is, is because of the laws of indices! If you recall the laws of indices:

√ab = (ab)1/2 = a1/2 x b1/2 = √a x √b

√(a/b) = (a/b)1/2 = a1/2 / b1/2 = √a / √b

*Note: There is a special case in which √a x √a = √a2 = a or (√a)2 = √a2 = a, and the reason behind this is because: √a x √a = a1/2 x a1/2 = a(1/2)+(1/2) = a1 = a

### 3) Mathematical operations of surds

When we add and/or subtract surds, it is different from how we do it with normal integers. Instead, visualize the surd as an entity on its own, like how we add/subtract when we deal with algebra.

For example: Simplify √48 + √75 without using a calculator.

√48 + √75 = √ (16 x 3) + √ (25 x 3)

= √16 x √3 + √25 x √3

= 4√3 + 5√3

= 9√3

*Note: To visualize, you can treat the surd (in this case, √3) like an unknown, x, so 4√3 + 5√3 = 4x + 5x = 9x

· Multiplying surds

When we multiply surds, if we are multiplying surds with an integer, we will only deal with the coefficient outside of the square root of the surd. i.e. 5 x 4√2 = 20√2

If we are multiplying a surd with another surd, and provided that the surds are of the same nature, we will typically end up with a rational number. i.e. 3√3 x 2√3 = 6(3) = 18

If we are multiplying a surd with another surd of different nature, we will be dealing with both the coefficients outside of the square root of the surds, and within the square root of the surds as well. i.e. 6√5 x 8√7 = (6 x 8) √ (5 x 7) = 48√35

· Conjugate surds

When we solve for the product of two surds, which are irrational in nature, we will typically end up with an irrational number as well. However, there is a special case which the product of two surds will give us a rational number. These two special surds are

called conjugate surds.

Given that p, q and a are all rational numbers and a > 0,

Product of (p + q √a) and (p – q √a), which are the conjugate of each other, is a rational number

For example: (5 + 6√7) (5 - 6√7) = 25 - 30√7 + 30√7 – 42

= 25 – 42

= -17

Through the example above, can you identify the part which shows why the product of conjugate surds will always end up with a rational number?

It lies all the way back to a formula in the basics of algebra manipulation, where (x + y) (x – y) = x2 – y2! The reason why this works is because the terms that contains surds will end up cancelling each other, and in addition to that, the terms that ends up multiplying another term with surds will end up being a rational number, because of its similar nature! (Recall the multiplication of surds with similar nature from the points above)

Therefore, the product of conjugate surds must be rational!

Question Time!

1. Without using a calculator, find the values of the integers x and y such that

[(a + b√5)/ (8 - 3√5)] = [(8 - 3√5)/ (3 - √5)]

2. Given that √ (p + q√7) = 4 / (3 - √7)2, where p and q are rational numbers, find the values of p and q.

3. Without using a calculator, find the value of a + b + c for which [(√12 - √6)/ (√15 + √12)] can be expressed as 2(√a + √b -2) - √c.

4. i) Express (3√5 – 4)2 in the form a + b√5.

ii) Hence, find the two square roots of 29 - 24√5.

5. Find the values of the integers a and b for which the solution of the equation x√5 = √27 - x√3 is (a + b√15)/2.

6. Given that 8 + 6√3 is a root of the equation x2 + ax + b = 0, where a and b are integers, find the value of a and b.

7. A right circular cylinder has a volume of (7 + 4√3) π cm3 and a base radius of (9 + √3) cm. Find its height in the form (p + q√3) cm, where p and q are rational numbers.

And that’s all for today, students! Math Lobby hopes that after this article, you have a clear understanding on the chapter of surds, and is equipped with the necessary skills to deal with questions involving the laws of surds and its mathematical operations!

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