Use exponent properties to transform exponential expressions and interpret growth/decay rates.
Determine whether two exponential expressions are equivalent by rewriting with exponent rules
Problem
Determine whether \(2^{t+3}\) and \(8\cdot~2^t\) are equivalent for every \(t\). Split the exponent sum, evaluate the constant power, and use the resulting common form to justify the conclusion.
Big Picture
What this problem is really about
Universal equivalence is established by rewriting both expressions into the same algebraic form, not by checking one input alone. We’ll split the exponent sum into a product of powers, evaluate the constant factor, compare coefficient, base, and exponent, and use a sample input only as a confirmation.
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