When [tex]$1,250^{\frac{3}{4}}$[/tex] is written in its simplest radical form, which value remains under the radical?
A. 2
B. 5
C. 6
D. 8
Answer (2)
Express 1250 as a product of its prime factors: 1250 = 2 × 5 4 .
Rewrite the expression: 125 0 4 3 = ( 2 × 5 4 ) 4 3 .
Apply the exponent and simplify: ( 2 × 5 4 ) 4 3 = 2 4 3 × 5 3 = 4 8 × 125 .
The value under the radical is 8 .
Explanation
Understanding the Problem We are given the expression 125 0 4 3 and asked to simplify it into radical form to identify the value under the radical.
Prime Factorization First, we express 1250 as a product of its prime factors. We have 1250 = 2 × 625 = 2 × 5 4 .
Rewriting the Expression Now we rewrite the original expression using the prime factorization: 125 0 4 3 = ( 2 × 5 4 ) 4 3 .
Applying the Exponent Next, we apply the exponent to each factor: ( 2 × 5 4 ) 4 3 = 2 4 3 × ( 5 4 ) 4 3 .
Simplifying the Second Factor We simplify the exponent of the second factor: ( 5 4 ) 4 3 = 5 4 × 4 3 = 5 3 = 125 .
Rewriting in Radical Form Now we rewrite the first factor in radical form: 2 4 3 = ( 2 3 ) 4 1 = 4 2 3 = 4 8 .
Final Simplification Putting it all together, we have 125 0 4 3 = 5 3 × 4 8 = 125 4 8 . The value that remains under the radical is 8.
Conclusion Therefore, when 125 0 4 3 is written in its simplest radical form, the value that remains under the radical is 8.
Examples
Understanding radical simplification is useful in various fields, such as engineering and physics, where complex calculations involving roots are common. For instance, when calculating the period of a pendulum, the formula involves a square root. Simplifying such expressions allows for easier computation and a better understanding of the physical quantities involved. Similarly, in electrical engineering, simplifying expressions with radicals can help in analyzing circuits and determining impedance values. These simplifications make complex problems more manageable and provide insights into the underlying principles.
When simplifying 125 0 4 3 , we find that the value remaining under the radical is 8. Therefore, the answer is option D. 8.
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