Which expression is equivalent to $\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}$
A. $\sqrt[16]{4^5}$
B. $\sqrt{2^5}$
C. 2
D. 4
Answer (1)
Multiply exponential terms in the numerator: 4 4 5 \tcdot 4 4 1 = 4 2 3 .
Divide exponential terms: 4 2 1 4 2 3 = 4 .
Raise the result to the power of 2 1 : ( 4 ) 2 1 = 2 .
The expression is equivalent to 2 .
Explanation
Understanding the Problem We are given the expression ( 4 2 1 4 4 5 \tcdot 4 4 1 ) 2 1 and asked to find an equivalent expression from the list: 16 4 5 , 2 5 , 2, 4.
Simplifying the Numerator Let's simplify the expression inside the parenthesis first. We know that when multiplying exponential terms with the same base, we add the exponents: a m \tcdot a n = a m + n . Therefore, 4 4 5 \tcdot 4 4 1 = 4 4 5 + 4 1 = 4 4 6 = 4 2 3 .
Simplifying Inside Parenthesis Now we have ( 4 2 1 4 2 3 ) 2 1 . When dividing exponential terms with the same base, we subtract the exponents: a n a m = a m − n . Therefore, 4 2 1 4 2 3 = 4 2 3 − 2 1 = 4 2 2 = 4 1 = 4 .
Final Simplification Now we have ( 4 ) 2 1 . Raising a number to the power of 2 1 is the same as taking the square root: a 2 1 = a . Therefore, ( 4 ) 2 1 = 4 = 2 .
Conclusion Therefore, the expression ( 4 2 1 4 4 5 \tcdot 4 4 1 ) 2 1 is equivalent to 2.
Examples
Understanding exponential expressions is crucial in various fields, such as calculating compound interest. For instance, if you invest money in an account with continuously compounded interest, the formula involves exponential terms. Simplifying these expressions helps you determine the future value of your investment more efficiently. Also, exponential expressions are used in physics to describe radioactive decay, where simplifying them allows for easier calculation of the remaining amount of a substance over time.
