156E - Indices and Surds

Indices

Indices are used to describe the general term for ² in say x². There are a few laws to know when manipulating expressions involving indices.

Examples

Laws of Indices

Surds

Surds are basically an expression involving a root, squared or cubed etc...

There are some basic rules when dealing with surds

Also notice the special case

Difference of Two Squares

 

 This is called the difference of two squares

Rationalising Surds

When you have a fraction where both the nominator and denominator are surds, rationalising the surd is the process of getting rid of the surd on the denominator.

To rationalise a surd you multiply top and bottom by fraction that equals one. Take the example shown below

1

√2 

To rationalise this multiply by effectively 1

1  × √2

――――

√2    √2

Can you see why √2⁄ √2 was chosen? This is because √2 ×√2  = 2 so the denominator becomes surd free.

1+√2 

―――

1+√5

For a more complex term

First of all, we need to get rid of the surd expression on the bottom; you should remember the difference of two squares formula.

So to get rid of the denominator surd we multiply (1+√2) ⁄ (1 - √5) by (1+√5) ⁄ (1+√5) like so.

In general 

FINDING LAST DIGIT IN A^B

In this post we will learn how to solve problems based on exponents i.e A^B. In exams generally questions based on finding the last digit in expression A^B are asked. So let us start learning this topic. This topic is useful for  CAT/SSC/BANK PO and various other competitive exams.

Let us take an example for better understanding.

Q. Find the last digit in 2347⁴¹?

A. As from sight this question appears to be very typical but if we go by the rules it is very easy to solve such type of problems. In this question we have to find the last digit in 2347⁴¹ for solving this type of problems we will only concentrate on the unit digit of the Number i.e in this case 7 and will find the last digit of 7⁴¹ not for the whole Number 2347⁴¹

Next part of solving this problem is based on Cyclicity of the number. Let us learn about cyclicity of a number.

Given a^b, units place digit of the result depends on units place digit of a and the divisibility of power b.

Consider powers of 2

As we know,

2¹ =2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128.. and so on

What do you observe here? We can see that the units place digit for powers of 2 repeat in an order: 2, 4, 8, 6. So the “cyclicity” of number 2 is 4 (that means the pattern repeats after 4 occurrences) and the cycle pattern is 2, 4, 8, 6. You can see from this  that to find the units-place digit of powers of 2, you have to divide the exponent by 4.

Let’s check the validity of above formula with an example.

Example: Find the units place digit of 2⁹⁹?

Using the above observation of cyclicity of powers of 2, first divide the exponent by 4. 99/4 gives reminder as 3. That means, units place digit of 2⁹⁹ is the 3rd item in the cycle which is 8.

Shortcuts to solve problems related to units place digit of a^b

Case 1: If b is a multiple of 4

·         If a is an even number, ie: 2, 4, 6 or 8 then the units place digit is 6

·         If a is an odd number, ie: 1, 3, 7 or 9 then the units place digit is 1

         

Case 2: If b is not a multiple of 4

·         Let r be the reminder when b is divided by 4, then units place of a^b will be equal to units place of a^r

CYCLICITY TABLE FOR YOU

Number	^1	^2	^3	^4	Cyclicity
2	2	4	8	6	4
3	3	9	7	1	4
4	4	6	4	6	2
5	5	5	5	5	1
6	6	6	6	6	1
7	7	9	3	1	4
8	8	4	2	6	4
9	9	1	9	1	2

So from the above table we get to know that the cyclicity of 7 is 4. So let us solve that question in which we had to find the unit digit of 2347⁴¹

And we have already discussed that we only need to find the unit digit of 7⁴¹ and cyclicity of 7 is 4 so divide 41 we will get remainder 1 so the unit digit in 7¹ is 7 hence the unit digit of expression 2347⁴¹ is 7.

In this way we can find the unit digit for any expression.

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