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
“X to the power of y,” or
“X to the y,” or
“yth power of X”
Examples: 23, 42, 165, 1031
Model (for explaining instances): repeated multiplication
Base: positive integers (e.g., 1, 2, 3, 4, 5)
Exponent: positive integers
Examples: 05, -24, 6-3, 00, -e1.8, (-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