Final answer:
The expression (4^5/4 \cdot 4^1/4 / 4^1/2) simplifies to 4 by using the properties of exponents, specifically by adding the exponents when multiplying and subtracting them when dividing.
Step-by-step explanation:
The expression in question is (4^5/4 \cdot 4^1/4 / 4^1/2). When working with exponents, we should remember that multiplying powers of the same base involves adding the exponents, and dividing them involves subtracting the exponents.
To simplify the given expression, we can use the properties of exponents:
- For multiplication: a^m \cdot a^n = a^(m+n)
- For division: a^m / a^n = a^(m-n)
Let's apply these rules:
- Firstly, add the exponents of the terms being multiplied: 4^(5/4 + 1/4) = 4^(6/4)
- Then, subtract the exponent of the denominator from the result of the first step: 4^(6/4 - 1/2) = 4^(6/4 - 2/4) = 4^1
- Thus, the expression simplifies to just 4, since any number to the power of 1 is itself.
Therefore, the expression (4^5/4 \cdot 4^1/4 / 4^1/2) is equivalent to 4.