B2.7 Evaluate and express repeated multiplication of whole numbers using exponential notation, in various contexts.
Skill: Evaluating and Expressing Repeated Multiplication of Whole Numbers Using Exponential Notation in Various Contexts.
Evaluating a power means determining the result. The power can be rewritten as a product to determine its result (for example, \({2^4}\; = \;2\; \times \;2\; \times \;2\; = \;16\)).
Powers are used to express very large and very small numbers. They are also used to describe very rapid growth (like doubling) that increases over time.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8 , 2020, Ontario Ministry of Education.
To contextualize the representation of repeated multiplication in exponential notation, it is good to identify real-life situations in which there is a number expressed as a power.
Conversion from metric units is one of them. For example, in a kilometre there are \(10\; \times \;10\; \times \;10\) metres, or 103 metres. Since there are 1000 millimetres in a metre, there are \(10\; \times \;10\; \times \;10\; \times \;10\; \times \;10\; \times \;10\) millimetres in a kilometre, or 106 millimetres.
The evaluation of the growth of a bacterial culture is usually the doubling time. If a bacterium doubles every 3 hours, in 12 hours there will be \(1\; \times \;2\; \times \;2\; \times \;2\; \times \;2\) bacteria, or \(\ 1 \times 2^4\) or 16 bacteria.
Knowledge: Exponential Notation
Exponentiation is a 5th operation, after addition, subtraction, multiplication and division.
Exponentiation is repeated multiplication and means "to raise a base to an exponent".
- 52 has a base of 5, an exponent of 2 and means \(5 \times \;5\) or 25;
- 105 has a base of 10, an exponent of 5 and means \(10 \times 10 \times 10 \times 10\) or 100 000.
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
It may be beneficial for the student's understanding to compare 23 and 32.
The expression 23 is expressed as 2 exponent 3, or 2 to the power of 3, 2 being the base and 3 being the exponent. The evaluation of this expression is 8.
\(2^3\; = \;2\; \times \;2\; \times \;2\; \; = \;4\; \times \;2\; = \;8\)
The expression 32 is expressed as 3 to the exponent 2, or 3 to the power 2, where 3 is the base and 2 is the exponent. The evaluation of this expression is 9.
\({3^2}\; = \;3\; \times \;3\; = \;9\)