Instances and Non-instances

What's in and what's out? What is the scope of our explanation and illustrations in this blog?

Instances are what (examples, concepts) we consider to be in the scope, and non-instances are those outside the scope of our discourse. 

Our goal is to develop deep conceptual understanding and procedural fluency at exponents and multiplication of exponents. The table below provides examples of various exponential forms (powers of) in the instances and non-instances.

Scope: Multiplication of exponents

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Power (an expression of): X^y

base=X            

exponent=y

pronounced as:

“X to the power of y,” or

“X to the y,” or

“y^th power of X”

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Instances:

Examples: 2³, 4², 16⁵, 103¹                                                                                                                                                                                                                                                                                                                                                                              

Model (for explaining instances): repeated multiplication 

Base: positive integers (e.g., 1, 2, 3, 4, 5)

Exponent: positive integers

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Non-instances:

Examples: 0⁵, -2⁴, 6^-3, 0⁰, -e^1.8, (-8/9)^1/3 , 0.56-^1/3

Model (for explaining non-instances): 

repeated division, ...

Base: variables, real numbers (e.g., -1, -22, -1/2, -5/9, 73/4, 3.142, p, 0.7, e, √2)

Exponent: negative integers, fractional, decimals