Final answer:
Doubling the exponent in an expression like 2n does not double the value of the expression; it results in exponential growth. So, I disagree with Miguel, because doubling the exponent increases the magnitude of the number at an exponential rate, not just doubling it.
Step-by-step explanation:
The expression 2n represents an exponential function where the base 2 is raised to the power of n. When Miguel suggests doubling the value of n, he is not actually doubling the value of the entire expression but rather increasing the exponent. According to the laws of exponents, if you double the exponent, you do not merely double the value of the expression, you square the value of the original base raised to that exponent. For instance, if n is 3, then 23 is 8. If we double n to 6, 26 becomes 64, which is the original value squared (82).
Exponential growth follows this pattern; it multiplies the base by itself a number of times equal to the exponent. So, in doubling sequences, we have the base raised to the number of doubling times. For example, after 5 doubling times starting at 1, we would have 25 equals 32. A key point to remember is additionally that as the exponent increases, the magnitude of the number (with a base greater than 1) also increases exponentially, not linearly.
When we talk about squaring of exponentials, for a term like (2n)2, we would square the base as usual, which gives us 4, and multiply the exponent by 2, leading to a final result of 22n which is different from the operation of simply doubling n in the expression 2n.