Which expression is equivalent to $24^{\frac{1}{3}}$ ?
A. $2 \sqrt{3}$
B. $2 \sqrt[3]{3}$
C. $2 \sqrt{6}$
D. $2 \sqrt[3]{6}$
Answer (1)
Find the prime factorization of 24: 24 = 2 3 × 3 .
Rewrite the cube root: 3 24 = 3 2 3 × 3 .
Simplify the cube root: 3 2 3 × 3 = 2 3 3 .
The equivalent expression is 2 3 3 .
Explanation
Understanding the Problem We are given the expression 2 4 3 1 and four options: 2 3 2 3 3 2 6 2 3 6 We want to find the expression that is equivalent to 2 4 3 1 . The expression 2 4 3 1 can be written as 3 24 .
Prime Factorization We need to simplify the expression 3 24 . To do this, we first find the prime factorization of 24. 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2 3 × 3 .
Rewriting the Cube Root Now we can rewrite the cube root using the prime factorization: 3 24 = 3 2 3 × 3
Simplifying the Cube Root Using the property of radicals, we can separate the cube root: 3 2 3 × 3 = 3 2 3 × 3 3 = 2 × 3 3 = 2 3 3
Final Answer Comparing the simplified expression 2 3 3 with the given options, we see that it matches the second option. Therefore, the expression equivalent to 2 4 3 1 is 2 3 3 .
Examples
Cube roots are useful in various fields, such as engineering and physics, when dealing with volumes. For example, if you have a cube with a volume of 24 cubic units, finding the length of one side involves calculating the cube root of 24, which simplifies to 2 3 3 . This concept is also applicable in financial calculations, such as determining the interest rate needed to triple an investment over a specific period.
