Evaluate $64^{\frac{1}{3}}$
Identify the radicand of $\sqrt[6]{4^8}$.
Answer (1)
Evaluate the expression 6 4 3 1 which is equal to 4.
Identify the radicand of 6 4 8 , which is the expression under the radical, 4 8 .
The final answer is: 4 ; 4 8
Explanation
Understanding the Problem We are asked to evaluate 6 4 3 1 and identify the radicand of 6 4 8 .
Evaluating the Expression To evaluate 6 4 3 1 , we need to find a number that, when raised to the power of 3, equals 64. In other words, we are looking for the cube root of 64, which is 3 64 . We know that 4 × 4 × 4 = 4 3 = 64 , so 3 64 = 4 .
Identifying the Radicand The radicand of a radical expression is the expression under the radical symbol. In the expression 6 4 8 , the radicand is 4 8 .
Final Answer Therefore, 6 4 3 1 = 4 and the radicand of 6 4 8 is 4 8 .
Examples
Understanding exponents and radicals is crucial in various fields, such as engineering and physics. For instance, when calculating the volume of a cube with side length s , the volume V is given by V = s 3 . Conversely, if you know the volume and need to find the side length, you would use the cube root, s = 3 V . Similarly, radicals are used in calculating distances and areas in geometry, making these concepts fundamental in practical applications.
