Indices

Before students learn the concept of indices they must have the knowledge on (i) factors and (ii) multiples. At later development on indices they have to have knowledge on (iii) like terms (iv) variables (v) algebraic expression.

When a number a is multiplied by itself a number of times, the product is called a power of a. For example, a × a is the second product of a; this is written in index form as a². The number that shows the power is called the index or exponent. The index of a² is 2 and the index of a⁵ is 5. The expression a⁵ means a×a×a×a×a. In a⁶, a is called the base.

Laws of Indices:

v  1^st Law [The Law of Multiplication ]

You know that 2³ = 2 × 2 × 2 and 2⁴ = 2 × 2 × 2 × 2.

     Therefore, 2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2)

                                         = 2 × 2 × 2 × 2 × 2 × 2 × 2

                                         = 2⁷

The above information shows one of the basic laws of indices (multiplication law of indices). This law can be written as; 

1.           an × am = an + m

Note that in both aⁿ and a^m, the base is the same. That is ‘a’.

Now exercise under this law:

Simplify the following and write your results in an exponential form.

1.     3³ × 3⁶

2.     2⁴ × 2⁴

3.     19¹² × 19¹³

Solution

1.      You know that 3³ = 3 × 3 × 3 and 3⁶ = 3 ×3 × 3 × 3 × 3 × 3.

Therefore, 3³ × 3⁶ = 3 ×3 × 3 × 3 × 3 × 3 × 3 × 3 × 3

                            = 3⁹.

Alternatively, that is applying the law we have:

We know that aⁿ × a^m = a^n+m, where a = 3, n = 3 and m = 6.

 Then 3³ × 3⁶ = 3³⁺⁶

           = 3⁹

Now work on your own question number (2) and (3). You can submit it through the contact me.

v  2^nd Law [ The Law of Division ]

The above information shows another basic law of indices (division law of indices). This law can be written as;

Note that in both aⁿ and a^m, the base is the same. That is ‘a’.

Now follow the explanation above and simplify the following and submit it to me through the contact me:

1)    9⁸ ÷ 9⁶

2)    3¹⁰ ÷ 3² ÷ 3³

3)    (2.5)⁴ ÷ (2.5)²

v  3^rd Law [ The Law of Exponent ]

You remember that when you work with brackets,

(2³)³ = 2³ × 2³ × 2³

From the first law, you can work out:

2³ × 2³ × 2³ = 2^{3 + 3 + 3}

                    = 2⁹.

Therefore, (2³)³ = 2⁹

The above information shows another basic law of indices (division law of indices). This law can be written as;

Now follow the explanation above and simplify the following and submit it to me through the contact me:

1.     16 × (25 ÷ 64)

2.     (5³)²

Note this very carefully!

We have now learnt the three fundamental laws or rules of indices. Using a, n and m these laws are:

Ø aⁿ × a^m = a^{n + m} – The Law of Multiplication

Ø aⁿ ÷ a^m = a^{n – m} – The Law  of Division

Ø (aⁿ)^m = a^nm – Law of Exponent

There is no fast rule for addition and subtraction other than what is being studied under algebra. Thus:

a^m + aⁿ = a^m + aⁿ and a^m - aⁿ = a^m - aⁿ since they are unlike terms.

You may also like these:

1.   Modular Arithmetic - An introduction - New!

2.  Negative Indices, Zero Indices and Exponents that are Fractions: - New!

3.   Operations in Modular Arithmetic (Addition, Subtraction, Multiplication and Equations.) - New!

4.  The Venn diagram – Two-Set Problem. - New!

 


For any further advice on your studies or any challenge on this post, please feel free to contact me at: dzaazo@gmail.com. I will appreciate it.