The 42nd Period | STEM Education Hub

Use this as a quick reference for the product property, the power property, negative exponents, and unit fraction exponents.

Exponential Function Manipulation infographic

🧭 Plot Summary

Lesson 2.3 gave you the general form f(x) = ab^x. Now we rewrite exponential expressions in equivalent forms so that the information you need is easy to see. The product property turns a horizontal translation into a vertical dilation. The power property turns a horizontal dilation into a change of base. The negative exponent property turns a negative exponent into a reciprocal, and unit fraction exponents are roots. Together they let you change the time unit of a model, for example from "doubles every 3 days" to "grows by about 26% each day."

What you'll do in this lesson

Use the product property to turn a horizontal translation into a vertical dilation: b^{x + k} = b^k·b^x.
Use the power property to turn a horizontal dilation into a new base: b^cx = (b^c)^x.
Rewrite negative exponents as reciprocals: b⁻ⁿ = 1 / bⁿ.
Evaluate unit fraction exponents as roots: b^1/k is the kth root of b, when it exists.
Combine the properties to change the time unit of a model, such as from every 3 days to every day.

Why it matters

Models are often given in one form but needed in another. A population that doubles every 3 days is written 2^t/3, but a daily growth factor is easier to compare with a percent change. Rewriting is how you move between the two without changing the function. Topic 2.5 uses these equivalent forms to build and interpret exponential models.

✅ Self-Check Before You Move On

Check off each item as you get there. These aren't grades — they're your own signal.

I can use the product property b^m·bⁿ = b^{m + n} to rewrite a function like 2^{x + 3} as 8·2^x.I can explain that every horizontal translation of an exponential function is equivalent to a vertical dilation, f(x) = b^{x + k} = ab^x with a = b^k.I can use the power property (bⁿ)^m = b^mn to rewrite a function like 2^3x as 8^x.I can explain that every horizontal dilation b^cx is equivalent to a change of base, (b^c)^x, for c ≠ 0.I can use the negative exponent property b⁻ⁿ = 1 / bⁿ, and I know a negative exponent does not make a number negative.I can evaluate an exponential unit fraction b^1/k as the kth root of b, when it exists.I can choose the equivalent form of an exponential expression that is most useful, such as the initial value, the daily factor, or the weekly factor.

🔬 Deep Dive

Your textbook for this topic: the product property, the power property, negative exponents, and unit fraction exponents — with interactive tools.

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🧭 Guided Practice

Step through worked examples for every essential knowledge statement in 2.4.A. Three examples each, revealed one step at a time.

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📝 Assignment

20 questions covering the whole lesson. Answer each one and get feedback right away, including the most likely reason behind a wrong choice.

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📌 AP Exam Note

Expect to rewrite an exponential expression into an equivalent form that reveals something useful: an initial value, a new base, or a growth factor per unit of time. Know which property does which job. A shift x + k becomes a multiplier b^k, a dilation cx becomes a base b^c, and a unit fraction exponent is a root. Always check your rewritten form by testing one input in both versions.