Exponential Function Manipulation
Three exponent rules run this topic: multiplying powers adds exponents, a power of a power multiplies them, and negative or fractional exponents mean reciprocals and roots. Read backward, the product rule regroups a·b to the power x plus c into a new coefficient times b to the x, which is why shifting an exponential horizontally is indistinguishable from stretching it vertically, and why a weekly growth factor hides a daily one inside it as a seventh root.
Every rule has a popular wrong version: bases multiplied when exponents should add, exponents multiplied when they should add, roots replaced by division, negative exponents read as negative numbers. The lesson states each rule exactly, shows the wrong versions failing on small numbers, and drills the rewrites the exam actually asks for.
The work
3 ways in · any order
Lesson
Exponential Function Manipulation
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The product, power, and negative-exponent properties used exactly, the regrouping that turns a horizontal shift into a vertical stretch, and time-scale conversions that take roots of factors instead of dividing them. Ten scenarios close it out, one popular wrong rule at a time.
Diagnostic
10-item topic check
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Ten items on the Topic 2.4 misconception family: exponent properties misapplied, bases multiplied, exponents added in the wrong place, roots turned into division. The bank ships with the Unit 2 data drop.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.