Use exponent properties to interpret exponential expressions, including percent growth and decay.
Validate an exponential rewrite with a rule and check
Problem
Evaluate the claim \(2^{t+3}=8\cdot~2^t\). Apply the product-of-powers rule to split the exponent sum, simplify \(2^{3}\), and compare both sides at \(t=1\) before stating whether the rewrite is valid.
Big Picture
What this problem is really about
A valid exponential rewrite needs a rule that proves equality for every input, not just a matching example. We’ll identify the exponent operation, apply the corresponding identity symbolically, simplify the fixed factor, and use zero and a nonzero input as arithmetic checks.
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