“…motivation comes from seeing the applicability of mathematics…”
When a group of University teachers of Complex Algebra and differential equations were asked,
“Why do we need to learn all these complex mathematical techniques if we may never use it for work?” They replied; “The purpose of learning Maths is not the final application in your daily life. But to develop analytical skills in your brain, and those skills are very useful for many job positions”
This article is primarily aimed at Engineers, Math students or enthusiasts who wish to gain a better understanding of Algebraic techniques such as the Laws of Indices and how it is practically applied in real life to solve engineering problems. At the end of the article, readers are expected to be able to explain basic concepts behind the Laws of indices with practical applications in everyday life activities such as the World Cup.
Keywords: Index, Power, Exponent, Indices, Algebra
Many engineering problems cannot be analysed without Algebraic techniques. Thus, intelligent application of Algebra is very crucial for solving complex problems and for mastering complicated relationships between modern engineering systems and products. Growing up in college, students constantly asked how they would ever apply most of the abstract Algebra concepts learned. In some cases, instructors shared similar concerns. There are more ridiculous memes and jokes how complex and abstract Math concepts are and/or how they ever get applied in real life aside reading and working them out in exercise books.
Motivated by such memes as above, this article has been written to concisely demonstrate practical applications of Laws of Indices as one of the many Algebraic Techniques taught in schools. The examples will also depict the essence of these calculation to develop analytical thinking skills. These applications are geared towards most common everyday life activities such as FIFA World Cup Fixtures. For some readers, this may be a revision, but I bet you have never used the Laws of Indices to understand World Cup Fixtures. And you will if you click here to the practical demonstration article.
There are mainly three laws of indices. An exponent or power or an index, is often used to write product of numbers in a much compact way. Consider the case of multiplying 3 by itself seven times. Basically you may see it in this form;
3 × 3 × 3 × 3 × 3 × 3 × 3
The above expression can as well be written more compactly as below;
³⁷ This is commonly read as ‘3 to the power 7’ or ‘3 exponent 7’
‘3’ can be referred to as the Base. ‘7’ is the Index, exponent or Power.
Law One
The first Law of Indices says, to find the product of two or more indexed expressions with the same base, the output is the common base to all the expressions exponent the sum of all the powers of involved expressions. For instance,
This can be verified by evaluating both sides separately.
Example Z⁴ × Z³ = Z^(4+3) = Z⁷
a¹ = a
Given the expression ; 7(⁷³) = ⁷¹ × ⁷³= 7^(1+3) = ⁷⁴
We can now verify that ⁷¹ = 7
Law Two
To divide two or more indexed expressions with the same base, the output is the common base to all the expressions exponent the subtraction of denominator exponent from the numerator exponent of involved expressions. For instance
a⁰ = 1
Example; ³⁴ /³⁴ = 1
But ³⁴/ ³⁴=3^(4–4) =³⁰
Thus,³⁰ = 1
Law Three
To simplify an indexed expression with another power, we multiply both powers to be come the only index of the base.
To give you time for a cup of coffee, you may click here to access how the Laws of Indices have been effectively applied to designing FIFA World Cup Fixtures. The case of RUSSIA 2018 World Cup has been used as an example.
