When $1,250^{\frac{3}{4}}$ is written in its simplest radical form, which value remains under the radical?
A. 2
B. 5
C. 6
D. 8
Answer (1)
Find the prime factorization of 1250: 1250 = 2 × 5 4 .
Rewrite the expression: 125 0 4 3 = ( 2 × 5 4 ) 4 3 .
Apply the exponent: ( 2 × 5 4 ) 4 3 = 2 4 3 × 5 3 .
Simplify to radical form and identify the value under the radical: 125 4 8 , so the answer is 8 .
Explanation
Understanding the Problem We are given the expression 125 0 4 3 and we want to simplify it to its simplest radical form to determine which value remains under the radical.
Prime Factorization of 1250 First, we need to find the prime factorization of 1250. We can write 1250 = 125 × 10 = 5 3 × 2 × 5 = 2 × 5 4 .
Rewriting the Expression Now we can rewrite the given expression using the prime factorization: 125 0 4 3 = ( 2 × 5 4 ) 4 3
Applying the Exponent Next, we apply the exponent 4 3 to each factor: ( 2 × 5 4 ) 4 3 = 2 4 3 × ( 5 4 ) 4 3 = 2 4 3 × 5 4 × 4 3 = 2 4 3 × 5 3
Simplifying to Radical Form We can rewrite 2 4 3 in radical form: 2 4 3 = 4 2 3 = 4 8 So, the expression becomes: 5 3 × 4 8 = 125 4 8
Identifying the Value Under the Radical The expression in simplest radical form is 125 4 8 . The value that remains under the radical is 8.
Examples
Understanding radical forms is crucial in various fields, such as engineering and physics, where complex calculations often involve simplifying expressions to make them easier to work with. For example, when calculating the impedance of an electrical circuit, you might encounter expressions involving radicals. Simplifying these expressions allows engineers to quickly determine the circuit's behavior and optimize its performance. Similarly, in physics, simplifying radical expressions can help in solving problems related to wave mechanics or quantum mechanics, where complex equations often arise.
