What is the simplified value of the exponential expression $16^{\frac{1}{4}}$?
A. $\frac{1}{2}$
B. $\frac{1}{4}$
C. 2
D. 4
Answer (1)
Rewrite 16 as a power of 2: 16 = 2 4 .
Apply the power of a power rule: ( 2 4 ) 4 1 = 2 4 ⋅ 4 1 .
Simplify the exponent: 4 ⋅ 4 1 = 1 .
The simplified value is 2 1 = 2 , so the final answer is 2 .
Explanation
Understanding the problem We are asked to find the simplified value of the exponential expression 1 6 4 1 . This means we are looking for a number that, when raised to the power of 4, equals 16.
Rewriting the expression We can rewrite 16 as 2 4 because 2 × 2 × 2 × 2 = 16 . So, we have 1 6 4 1 = ( 2 4 ) 4 1 .
Applying the power of a power rule Using the power of a power rule, which states that ( a m ) n = a m ⋅ n , we can simplify the expression further: ( 2 4 ) 4 1 = 2 4 ⋅ 4 1 .
Simplifying the exponent Now, we simplify the exponent: 4 ⋅ 4 1 = 1 . Therefore, the expression becomes 2 1 .
Final Answer Finally, 2 1 = 2 . So, the simplified value of the exponential expression 1 6 4 1 is 2.
Examples
Imagine you are calculating the side length of a square garden. If the area of the garden is 16 square meters, and you want to find the length of one side, you would take the square root of 16, which is 1 6 2 1 = 4 meters. Now, suppose you have a four-dimensional hypercube with a 'hypervolume' of 16 units, and you want to find the length of one of its sides. In this case, you would calculate the fourth root of 16, which is 1 6 4 1 = 2 . This concept extends to various fields, including engineering and physics, where understanding roots and exponents is crucial for solving complex problems involving dimensions and scaling.
