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
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
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 (
In general
FINDING LAST DIGIT IN AB
In this post we will learn how to
solve problems based on exponents i.e AB. In exams
generally questions based on finding the last digit in expression AB 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 ab,
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
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 ab
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 ab will
be equal to units place of ar
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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