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: âˆš2, âˆš7, âˆš139

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 âˆš4, âˆš16 and âˆš25 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: âˆš4 = 2, âˆš16 = 4, âˆš25 = 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: âˆšab = âˆša x âˆšb,

Law 2: âˆš(a/b) = âˆša / âˆšb,

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:

âˆšab = (ab)1/2 = a1/2 x b1/2 = âˆša x âˆšb

âˆš(a/b) = (a/b)1/2 = a1/2 / b1/2 = âˆša / âˆšb

*Note: There is a special case in which âˆša x âˆša = âˆša2 = a or (âˆša)2 = âˆša2 = a, and the reason behind this is because: âˆša x âˆša = a1/2 x a1/2 = a(1/2)+(1/2) = a1 = a

### 3) Mathematical operations of surds

Â· Addition/Subtraction 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 âˆš48 + âˆš75 without using a calculator.

âˆš48 + âˆš75 = âˆš (16 x 3) + âˆš (25 x 3)

= âˆš16 x âˆš3 + âˆš25 x âˆš3

= 4âˆš3 + 5âˆš3

= 9âˆš3

*Note: To visualize, you can treat the surd (in this case, âˆš3) like an unknown, x, so 4âˆš3 + 5âˆš3 = 4x + 5x = 9x

Â· 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 4âˆš2 = 20âˆš2

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. 3âˆš3 x 2âˆš3 = 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. 6âˆš5 x 8âˆš7 = (6 x 8) âˆš (5 x 7) = 48âˆš35

Â· 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 âˆša) and (p â€“ q âˆša), which are the conjugate of each other, is a rational number

For example: (5 + 6âˆš7) (5 - 6âˆš7) = 25 - 30âˆš7 + 30âˆš7 â€“ 42

= 25 â€“ 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 â€“ y) = x2 â€“ 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 + bâˆš5)/ (8 - 3âˆš5)] = [(8 - 3âˆš5)/ (3 - âˆš5)]

2. Given that âˆš (p + qâˆš7) = 4 / (3 - âˆš7)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 [(âˆš12 - âˆš6)/ (âˆš15 + âˆš12)] can be expressed as 2(âˆša + âˆšb -2) - âˆšc.

4. i) Express (3âˆš5 â€“ 4)2 in the form a + bâˆš5.

ii) Hence, find the two square roots of 29 - 24âˆš5.

5. Find the values of the integers a and b for which the solution of the equation xâˆš5 = âˆš27 - xâˆš3 is (a + bâˆš15)/2.

6. Given that 8 + 6âˆš3 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 + 4âˆš3) Ï€ cm3 and a base radius of (9 + âˆš3) cm. Find its height in the form (p + qâˆš3) cm, where p and q are rational numbers.

And thatâ€™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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