We saw in index notation in Pre-Algebra on how to represent numbers using basic index notation. We’ll now look at how we can use them to execute various mathematical calculations, such as multiplication and division.
When you multiply two pronumerals together, or multiply a pronumeral to itself by a certain number of times, we write the expression in an index form for the repeated pronumeral. This representation is called index notation.
We know y x y is y^{2}. The 2 in y^{2} is called the power or index, and y is the base.
Similarly p x p x p x p = p^{4}.
Here p is the base, and 4 is the index (or power) of p. Expressing numbers with a base and an index is called index notation.
We’ll now look at a number of index laws
Index law for multiplication
This law states that when you have two index numbers to multiply
- multiply any co-efficients first, and then
- keep the same base, and add the indices
In other words, x^{n} x x^{m} = x^{n+m}, and ax^{n} x bx^{m} = abx^{n+m}
Example 1: 2^{3} x 2^{4}
= (2 x 2 x 2) x (2 x 2 x 2 x 2)
= 2^{7}
Example 2: a^{3} x a^{5}
= (a x a x a) x (a x a x a x a x a)
= a^{8}
Example 3: 2a^{2} x 3a^{3}
= (2 x a x a) x (3 x a x a x a)
= (2 x 3) x (a x a x a x a x a)
= 6a^{5}
Index law for division
This is the opposite of index law for multiplication. When you divide two index numbers, you should obey the following rules:
- divide any co-efficients first, and then
- keep the same base, and subtract the indices
In other words, x^{n} ÷ x^{m} = x^{n-m}, and = x^{n-m}
Example 1: 2^{6} ÷ 2^{2}
=
= 2 x 2 x 2 x 2
= 2^{4}
Example 2: y^{7} ÷ y^{3}
=
=
= y^{4}
Example 3: 12a^{5} ÷ 4a^{4}
= x
= 3a
Index law for powers
To simplify an expression with further power of indices, you should obey the following rules:
- raise any co-efficient to the power outside the grouping symbols, and
- keep the same base, and multiply the indices
In other words, = x^{mn}, and = a^{n}.x^{mn}
Example 1:
= 2^{3} x 2^{3}
= 2^{3+3} (using the index law of multiplication)
= 2^{6}
= 2^{3×2} = 2^{6}
Example 2:
= a^{2} x a^{2} x a^{2}
= a^{2+2+2} (using the index law of multiplication)
= a^{6}
= a^{2×3} = a^{6}
Example 3:
= 2a^{2} x 2a^{2} x 2a^{2}
= (2 x 2 x 2) x (a^{2} x a^{2} x a^{2}) (using the index law of multiplication)
= 8 x a^{6}
= 8a^{6}
Index law for power 0
This is a special type of an index law for powers – how to calculate the expression involving an index (or power) of 0?
Using the index laws for division, we can see that = 1, since
x^{m-m} = x^{0} = 1
Hence we get x^{0} = 1, i.e. any number or expression to the power of zero is 1.
Note: 1. Another name for index is power or exponent.
2. The scientific calculator button for power is
Press base ‘number’ x, then the power button, then enter the ‘index’ y.
Summary of Index Laws & Index Notation
Index law for multiplication |
x^{n} x x^{m} = x^{n+m}, andax^{n} x bx^{m} = abx^{n+m} | |
Index law for division |
x^{n} ÷ x^{m} = x^{n-m}, and = x^{n-m} | |
Index law for powers |
= x^{mn}, and = a^{n}.x^{mn} | |
Index law for power 0 |
x^{0} = 1 |
In addition, there are two other laws of index – law for negative indices, and law for fractional indices