Final answer:
To write a function in the form of 3(3^(t+2)) using properties of exponents, one must apply the rule of raising a number to an exponential term by cubing the digit and multiplying the exponent term by 3, which conforms to the principle that (a^n)^m = a^(nm).
Step-by-step explanation:
The student is asking how to write a function in the form of 3(3t+2) using properties of exponents. In mathematics, when we are cubing exponentials, we need to cube the digit term in the usual way and multiply the exponent by 3. This is because when an exponential term like an is raised to a power m, the new exponent becomes the product of the original exponent and the power, or anm.
An example that illustrates this is when we have (53)4, this can be seen as multiplying the exponents 3 and 4 to get 12, and hence, we write it as 512. Similarly, for a term like 31.7, we can interpret this as taking the tenth root of 3 and raising it to the 17th power, following the principle of exponentiation.
To follow the rules of exponentiation properly, we must ensure that the base remains the same and only the exponents are manipulated when performing operations like multiplication and power raising. If we were to write an expression like 32 × 103 times 2 × 102, we distribute the powers accordingly and simplify as follows: 3.2 × 2 × 103 × 102 = 6.4 × 105.