Rules for Fractions and Indices
Fractions
A fraction represents a part of a whole number or collection of objects.
It consists of a numerator (top number) and a line that separates it
from a denominator (bottom number).
Addition and Subtraction of Fractions
When adding or subtracting fractions the following rules apply:
1.
If the denominators of two or more fractions are the same, we
simply add or subtract the numerators of the fractions and keep
the denominator the same.
Examples:
1
4
1
2
4
4
4
5
+ =
1
3
5
5
− =
2.
If the denominators of two or more fractions are not the same,
we need to find the LCM (lowest common multiple) of the
denominators, and use that number as a common denominator.
We then divide the common denominator by the denominators
of the fractions that are being added or subtracted and then
multiply them by their corresponding numerators.
Example:
1
3
+
2
4
=
(4×1) + (3×2)
12
=
4+6
12
=
10
12
=
5
6
 In this example the common denominator is 12.
 The denominator of the first fraction, which is 3, is divided
into 12 and gives 4 as a result. The 4 is then multiplied by
the numerator of the first fraction, which is 1, and gives 4 as
the product.
 The denominator of the second fraction, which is 4, is
divided into 12 and gives 3 as a result. The 3 is then
multiplied by the numerator of the second fraction, which is
2, and gives 6 as the product.
 The two products (4 and 6) are then added to give the sum
of 10 and placed over the common denominator (12).
 The answer is therefore,
𝟏𝟎
𝟏𝟐
and reduced to
𝟓
𝟔
by dividing
the numerator and denominator by 2.
Example:
7
8
−
3
4
=
(1×7) − (2×3)
8
=
7−6
8
=
1
8
 This example follows the same steps as the previous one
done for addition except that subtraction of the two
products (7 and 6) is done instead of addition.
Multiplication of Fractions
Two or more fractions can be multiplied together regardless of their
denominators being the same or different.
The process is to simply multiply all numerators together and all
denominators together of the respective fractions.
The answer will be the product of the numerators above the product of
the denominators.
Example:
4
5
×
3
7
=
4×3
5×7
=
12
35
Division of Fractions
To divide one fraction by another, the second fraction must be
inverted, which means to switch the numerator with the denominator.
Multiplication of the first fraction with the inverted second fraction can
then take place to find the quotient (result) of the two fractions.
Example:
4
5
3
4
7
5
÷ =
7
4×7
3
5×3
× =
=
28
15
= 1
13
15
 From the above example, the second fraction
inverted to
7
3
.
3
7
has been
Indices (plural of index)
An index or power of a number represents the number of times to use
the number in multiplication.
The index is written as a small number to the top right of a bigger
number, which is known as the base.
INDEX or POWER
42
BASE
42 = 4 × 4 = 16
Multiplication of Indices
When multiplying indices with the same base, the indices are added.
Example:
42 × 42 = 42+2 = 44 = 256
Division of Indices
When dividing indices with the same base, the indices are subtracted.
Example:
35 ÷ 32 = 35−2 = 33 = 27
Brackets with Indices
When there is an index outside the bracket, the indices are multiplied.
Example:
(23 )2 = 23×2 = 26 = 64
Power of Zero
Any number that has an index of zero is equal to 1
Examples:
40 = 1
100 = 1
500 = 1
1200 = 1
Negative Indices
When an index is negative, we take the base and put 1 over it, changing
the index positive in the process.
Example:
6−2 =
1
1
6
36
2 =
Fractional Indices
When the index is a fraction, the denominator is the root of the base.
We then raise the answer to the power of the numerator.
Example:
3
2
82/3 = ( √8) = (2)2 = 22 = 4