GEEM - Skills and Knowledge

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.

To evaluate a power means to determine the result. Often the power would 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 (such as 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 fifth 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 multiplied by itself 2 times, $5 \times 5$ or 25;
10⁵ has a base of 10, an exponent of 5 and means 10 multiplied by itself 5 times $10 \times 10 \times 10 \times 10 \times 10$ or 100 000.

Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8 , 2020, Ontario Ministry of Education.

It may be beneficial for the student's understanding to compare 2³ and 3².

The expression 2³ is expressed as 2 to the exponent 3, or 2 to the power of 3, where 2 is 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.

$3^{2} = 3 \times 3 = 9$