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A-SSE.3.c Warmup M2-013-A08-V03

Use exponent properties to transform exponential expressions and interpret growth/decay rates.

Compare exponential models across different time units using \((b^k)^t = b^{kt}\)

Problem

Compare \(N(t)=200\cdot~2^t\) and \(N(t)=25\cdot~2^{t+3}\) for the same input \(t\). Split the shifted exponent, calculate the constant factor, and decide whether the resulting expressions agree for every \(t\).

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They are not equivalent because the second model has factor 2³ = 8 per interval. The exponent t still gives the per-interval factor 2.

They are equivalent for the same t. Both have interval factor 2, and 25 · 2t ⁺ ³ = 25 · 2³ · 2t = 200 · 2t.

They are equivalent because both initial values are 200. No coefficient algebra establishes that claim.

They are not equivalent because their visible coefficients are 200 and 25. The shifted exponent contributes 2³, making 25 · 2³ = 200.

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Curriculum context

Course: Math II
Standard: A-SSE.3.c
Category: Algebra
Domain: Seeing Structure in Expressions
Objective: Use exponent properties to transform exponential expressions and interpret growth/decay rates.
Problem type: Compare exponential models across different time units using \((b^k)^t = b^{kt}\)