Surds - The basic part 3

Rationalizing the denominator  --> Simply means, NO surds in the denominator!

So how?

Type 1: Only a single surd term in the denominator

Examples of those with a single surd term in the denominator are

So how to rationalize them? Simple  -> multiply both the numerator and denominator by the surd you see in the denominator.

Type 2: Two terms in the denominator, with at least one of them being a surd.

So how to rationalize them? See example below:

Surds - The basic part 2

How to simplify surds?

Let's look at square root of 12. Think of 12 = 4 x 3. Hence:

Try these yourself:

Answers:

Surds -- The basic part 1

Square root simply means power half!

How about these?

Indices

Base vs Power

Remember the following rules:

Same base:

a^m x aⁿ = a ^(m+n)

a^m / aⁿ = a ^(m-n)

Same power

a^m x b^m = (ab) ^m

a^m / b^m = (a/b) ^m

Same base and same power then you can add or subtract:

2a^m + a^m =  a^m (2+1) =3a^m

2a^m - a^m =  a^m (2-1) =a^m

^{Solving indices equations type 1 --> There are only 2 non- zero terms, and they can be written into two with the same base.}

E.g. Solve 2^x = 8 ^x-3

Step 1: Are there 2 non- zero terms only?  Yes

Step 2: Can they be written into terms with the same base? Yes, as 8 can be written as 2³

Step 3: Write them into the same base, and compare the power.

2^x = 2^3(x-3)

x = 3(x-3)

x = 3x -3

x – 3x = -3

-2x = -3

x = 1.5

Solving indices equations type 2 --> There are only 2 non- zero terms, and they CANNOT be written into two with the same base.

2^x = 5 ^2x-3

Step 1: Are there 2 non- zero terms only?  Yes

Step 2: Can they be written into terms with the same base? No. 5 and 2 cannot be simplified to a number with the same base.

Step 3: ln both sides, and make x the power.

2^x = 5 ^2x-3

ln2^x = ln5 ^2x-3

xln2 = (2x-3)ln5

xln2 =2xln5 – 3ln5

2xln5+xln2 = 3ln5

x(2ln5 +ln2) = 3ln5

x = 3ln5/(2ln5+ln2) =1.23

Solving indices equations type 3 --> You can simplify to 3 non- zero terms. 2 of the terms have the same base, and one has twice the power of the other.

4^x -2.2^x +1 =0    ----- (1)

4^x  = 2^2x

Let y = 2^x, and so y² = 2^2x.

Equation (1) become:

y² -2y+1 =0

(y-1)² = 0

y=1, remember to substitute back y = 2^x, since you are solving for x.

2^x =1 à this is solving using type 1 method, as 1 = 2⁰.

2^x = 2⁰

x=0

Knowledge Check (Quadratic Equation)

Questions

1. Express in terms of k and h, the sum of root and product of roots of x² + kx +h.

2. Given 2x² + kx +8 is always positive, find the range of values of k.

3.  Complete the square for 2x² + 3x -4, express it in the form of a(x-b)² + c.

4. Sketch the curve y= x² + 3x -4, -5 ≤x≤ 3.

5. Find the range of values of 9 - x² > 0.

6. Find the range of values of x² + 5x - 6 < 0.

7. Given y =2x² + kx+ 8 intersects y= x at 2 distinct points. Find the range of values of k.

8. Given  y= x is tangent to y =2x² + kx+ 8, find the values of k.

9. Given  y= x is does not intersect  y =2x² + kx+ 8, find the range of values of k.

Answers

1	Sum of roots = -k; Product of roots = h.
2	Discriminant < 0 k2 - 4(8)(2)< 0 k2 - 64 < 0  (k + 8)(k-8) <0 -8 < k < 8
3	a = 2; b=3/4 ; c= -71/16
4	y= x2 + 3x -4, -5 ≤x≤ 3   = (x+4)(x-1) x -5 -4 -1.5 0 1 3 y 6 0 -6.25 -4 0 14
5	9 - x2 > 0 x2 -9 > 0  (x+ 3)(x-3) > 0 x> 3 or x< -3
6	x2 + 5x - 6<0 (x +6)(x-1) <0 -6 < x <1
7	y =2x2 + kx+ 8 ----- (1) y= x                 ------(2) 2x2 + kx+ 8 = x 2x2 +( k-1)x+ 8 =0 Apply discriminant > 0 ( k-1)2 - 4(2)(8)>0  k2 -2k+1-64 >0 k2 -2k-63 >0 (k-9)(k+7)>0 k> 9 or k<-7
8	y =2x2 + kx+ 8 ----- (1) y= x                 ------(2) 2x2 + kx+ 8 = x 2x2 +( k-1)x+ 8 =0 Apply discriminant =  0 ( k-1)2 - 4(2)(8)= 0  k2 -2k+1-64 = 0 k2 -2k-63 = 0 (k-9)(k+7)= 0 k= -7 or k = 9
9	y =2x2 + kx+ 8 ----- (1) y= x                 ------(2) 2x2 + kx+ 8 = x 2x2 +( k-1)x+ 8 =0 Apply discriminant <  0 ( k-1)2 - 4(2)(8)< 0  k2 -2k+1-64 < 0 k2 -2k-63 < 0 (k-9)(k+7)< 0 -7 <k< 9

How do you study & remember?

Many students comment:
- once they move on to the next chapter, they forget the previous.
- if they study in advance, they forget close to the exams.

So, the golden question. How to remember what you have studied?

Similarly, how do you remember 1+ 1= 2 till today? Did you practise that recently?

I believe, the best way to remember for a long period of time is to use it so often that you don't need to spend time memorise it. That is, after some time, you spend time recalling the work rather than memorising them.

I am a believer of "progressive" studying. If you need to study chapters 1, 2 and 3, do it this sequence:
- study chapter 1
- study chapter 1 + 2
- study chapter 1+ 2 + 3
- practice chapter 1+ 2+ 3 often
and when you have time, recall on the bus or when you have time, what you have studied.

I use to do that often, and after a decade, I still remember many of the subjects that I have studied.

I do not memorise them, I recall them when I teach. Of course you need to recall, else you will forget. =)
For subjects e.g. history and geography, which I did not bother recalling for the past decade, I truly forget.

Memorise & Recall. Then you will remember!

For maths, there is an additional step, practice.

Memorise, recall & practice!

All the best!