Final answer:
To simplify ((3/3)4(3/4)3)2, we first recognize (3/3)4 is 1 and (3/4)3 is 27/64. Applying the outer exponent, we square both numerator and denominator to get 729/4096 as the equivalent expression.
Step-by-step explanation:
The student is asking to simplify the mathematical expression ((3/3)4(3/4)3)2. This involves understanding the rules of exponents and how to apply them when dealing with numbers and variables in parentheses that are raised to a power. In this case, we must first simplify what is inside the parentheses before applying the outside exponent.
Inside the parentheses, we have (3/3)4 and (3/4)3. Since 3/3 is equal to 1, (3/3)4 simplifies to 1^4, which is still 1. On the other hand, (3/4)3 represents the number 3/4 being cubed, so we need to multiply 3/4 by itself three times. The resulting value is 27/64. Now we can rewrite our expression: (1 * 27/64), which is simply 27/64.
With the expression simplified inside the parentheses to 27/64, we can now apply the outside exponent of 2. By raising 27/64 to the second power, we square both the numerator and the denominator: (27^2)/(64^2). Squaring 27 gives us 729, and squaring 64 gives us 4096. Therefore, the final simplified expression is 729/4096.
It is crucial to remember the rules of cubing of exponentials, where you raise each number inside the parentheses by the power requested, as well as when dealing with numbers raised to an outside power, to raise both the numerator and denominator separately.