Evaluate: $64^{\frac{1}{3}}$
A) 4
B) 8
C) 2
D) 16
Answer (1)
Recognize that 6 4 3 1 means finding the cube root of 64.
Express 64 as 4 3 .
Simplify 3 4 3 to 4.
The final answer is 4 .
Explanation
Understanding the problem We are asked to evaluate 6 4 3 1 . This expression represents the cube root of 64.
Finding the cube root We need to find a number that, when multiplied by itself three times, equals 64. In other words, we are looking for a number x such that x 3 = 64 .
Prime factorization We can express 64 as a product of its prime factors. 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2 6 .
Simplifying the expression Now, we can rewrite 2 6 as ( 2 2 ) 3 = 4 3 . Therefore, 3 64 = 3 4 3 = 4 .
Final Answer Thus, 6 4 3 1 = 4 .
Examples
Imagine you're designing a cubic storage container and you want it to hold exactly 64 cubic feet of volume. To find the length of each side, you need to calculate the cube root of 64, which is 4 feet. This ensures your container has the desired volume. Understanding cube roots is essential in various fields like engineering, architecture, and even in everyday tasks like optimizing storage solutions.
