Final answer:
Upon simplifying the given complex expression, we find that it reduces to B) 2^(4t−7), which is the only option that matches the simplified form.
Step-by-step explanation:
To determine whether each expression is equivalent to [(16^t-0.25(4)^2t) / (4^0.5(16)^t)] ÷ [(2^4) / (16^t)], we need to simplify the given expression and compare it to the options A) 3 * 2^(4t−7) B) 2^(4t−7) C) 2^(4t−3) D) 3 * 2^(4t−3) E) 2^(4t−4).
First, we simplify the numerator and see that 0.25(4)^2t simplifies to (1/4)*4^2t which is 4^t. So the numerator becomes 16^t - 4^t. Noticing that 16 is 4^2 we can express 16^t as (4^2)^t which is 4^(2t). Therefore, the numerator is 4^(2t) - 4^t.
The denominator first part is 4^0.5 which is 2 and (16)^t is the same as 4^(2t), so the first denominator simplifies to 2*4^(2t). Now, looking at the second part of the overall denominator, 2^4 equals 16, so the denominator simplifies to 16^t.
Combining the two parts of the denominator yields (2*4^(2t)*16^t). This results in 4^(2t + 2t) which is 4^(4t). So the overall expression becomes (4^(2t) - 4^t) / 4^(4t).
Now we divide each term in the numerator by 4^(4t) which gives us 4^(-2t) - 4^(-3t). Since 4 equals 2^2, we can convert this to 2^(-4t) - 2^(-6t). When we take the common base, the expression simplifies to 2^(-6t) * (2^(2t) - 1). Knowing 2^(2t) is always greater than 1 for any t > 0, the expression can't be negative, so we can safely say that the options with a positive power of 2 are potential matches.
Reviewing the options, the only one that matches this form is option B) 2^(4t−7), because when rewritten, 2^(4t−7) becomes 2^(-6t)*2^(2t), which follows the form we derived correctly.