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What is an index and indices?
An index is a number that floats on the top right-hand side of a number. This can appear on the top right-hand side as a letter or a number.
An index is a singular form whereas multiple indexes make indices. Indices show how many times the letter or element is multiplied to its own.
For example-
22 is an example of indices, where two is two times as in two is multiplied two times by itself: 2 x 2 = 4.
This can also be read as ‘two raised to the power of two’ or ‘the square of two’.
23 is an example of indices, where two is three-time as is two is multiplied three times by itself: 2 x 2 x 2 = 8.
This can also be read as ‘two raised to the power of three’ or ‘the cube of two’.
X2 is an example of indices, where ‘X’ is two times as in ‘X’ is multiplied two times by itself: X x X = X2.
This can also be read as ‘X raised to the power of two’ or ‘the square of X’.
X3 is an example of indices, where ‘X’ is three-time as is ‘X’ is multiplied three times by itself: X x X x X = X3.
This can also be read as ‘X raised to the power of three’ or ‘the cube of X’.
The Laws of Indices
The laws of indices are certain rules or laws one must follow while solving algebra-based expressions involving indices.
x2 : In this expression ‘x’ is considered to be a base and ‘2’ is the ‘power’ or indices of ‘x’.
Law 1– Law of Multiplications.
If two elements in a given multiplication equation with elements ‘that have the same bases’ then their powers are added.
General representation of this law- ax x ay = ax+y
The equation has a common base all over that is ‘a’.
Examples of this given law:
x2x x3x x4
= x2+3+4
= x9.
22 x 22 x 21 x 210
= 22+2+1+10
= 215.
2a x 2b x 2-c x 2-d
= 2a+b-c-d.
Law 2– Law of Division.
If two elements in a given division equation with elements ‘that have the same bases’ then their powers are subtracted.
General representation of this law- ax / ay = ax-y
The equation has a common base all over that is ‘a’.
Examples of this given law:
X4 / x2 / x1
= x4-2-1
= x1
= x.
220 / 212 / 22 / 22
= 220-12-2-2
= 24.
2a / 2b / 2c / 2d
= 2a-b-c-d.
Law 3– Law of Brackets.
If an element already having power is again given another power then that new power is written with a bracket (outside the bracket) and both the powers are multiplied.
General representation of this law- (ax)y = axy.
It is fine if all the elements do not possess the same base. This law still is applicable.
Examples of this given law:
(22)2
= 22×2
= 24.
(220)2
= 220×2
= 240.
(x3)3
= x3×3
= x9.
(xa)b
= xaxb
= xab.
Law 4– Law of Negative Powers.
If a number or letter has a negative power, then the whole number is written as a fraction with a reciprocal along with the power. The power then becomes positive.
General representation of this law- a-x = 1/ax
Examples of this given law:
10-2
= 1/ 102
= 1/100.
2-10
= 1/ 210
= 1/1024.
a-1 + a-2
= 1/ a1 + 1/a2.
(7-a)b
= (7-ab)
= 1/7ab.
Law 5– Law of Power of Zero.
If a number or letter has zero as its power, then the answer for that is not zero but one.
General representation of this law- a0 = 1
Note that the answer for powers of zero will always be one and not zero. A number with the power of zero is not zero, it is one!
Examples of this given law:
200
=1.
100000
= 1.
(17-a)0
= 70
= 1.
22 x 27 x 21 x 2-10
= 22+7+1-10
= 210-10
= 20
= 1.
Law 6– Law of Power of Fractions.
Both the denominator and numerator in a fraction or element have a meaning.
The denominator of a fraction tells us the type of root:
For example- x1/2 indicates a square root √x, and the numerator of a fraction indicates the normal power for the whole element, x3/2 indicates (√x)3.
Examples of this given law:
y7/3
= (3√y)7
y7/3 x y2/3
= y7+2/3
= y9/3
= (3√y)9.
495/2
= (√49)5
= (7)5
= 7 x 7 x 7 x 7 x7
= 16807.
Conclusion
The rules of indices play a crucial role when it comes to time management while solving a mathematical equation and Laws of Indices questions with lots of calculations. Just like an ABC, one must know these laws of indices by heart for better and ease in the field of mathematics.
