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Investigating exponent rules
Write 64 in as many ways as possible using exponent notation. For instance 32*2 = 2^5*2^1 or 16*4=2^4*2^2 or 2*2*2*2*2*2=2^6 etc. As we know that these must all be the same number then we can see that 2^a*2^b=2^(a+b).
You can come up with the rule that a^c*b^c=(a*b)^c by coming up with the factorizations of 225 for instance.
You can also have students conjecture rules first and then use factorizations like these to check which rules work.
IBL Calculus linear and exponential exploration which helps students develop a characterization of liner and exponential growth: https://sites.google.com/site/aibldraftinstructorsite/home/in-class-activities-and-activity-guides/ca-i4-linear-and-exponential-characteristics
IBLCalculus Precalculus page contains a whole section on Exponential functions with a variety of activities: https://sites.google.com/site/aibldraftinstructorsite/precalculus
Use www.desmos.com to compare power functions and exponential functions. Have students graph 2^x and x^2 from 0 to infinity. When is 2^x bigger? When is x^2 bigger? In the long run which function is bigger? Do the same thing with 3^x and x^3?
Make table for (-2)^x. Notice in this Desmos graph that Desmos will graph the table of (-2)^x but not the form f(x)=(-2)^x - why is this? Is this a function? What is the domain? What is the range? https://www.desmos.com/calculator/iqyqr27hhu
Desmos Activities: https://teacher.desmos.com/exponential