Use exponent properties to transform exponential expressions and interpret growth/decay rates.
Rewrite an exponential expression to reveal the initial value at input 0
Problem
Rewrite \(5\cdot~2^{t+3}\) as \(a\cdot~b^t\) by splitting \(2^{t+3}\) into variable and constant powers. Calculate the new coefficient, state the value at \(t=0\), and verify it in the original expression.
Big Picture
What this problem is really about
A constant shift in the exponent contributes a constant power factor that can be absorbed into the coefficient. We’ll split the exponent sum, evaluate that fixed power, combine only the constant factors, and check input zero in both forms to verify the newly exposed initial value.
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