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 (2)
Express 1250 as a product of its prime factors: 1250 = 2 × 5 4 .
Substitute into the expression: ( 2 × 5 4 ) 4 3 = 2 4 3 × 5 3 .
Rewrite 2 4 3 in radical form: 4 2 3 = 4 8 .
The simplified expression is 125 4 8 , so 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 = 125 × 10 = 5 3 × 2 × 5 = 2 × 5 4 .
Applying Exponent Rules Now, substitute this into the original expression: 125 0 4 3 = ( 2 × 5 4 ) 4 3 . Using the properties of exponents, we get ( 2 × 5 4 ) 4 3 = 2 4 3 × ( 5 4 ) 4 3 = 2 4 3 × 5 4 × 4 3 = 2 4 3 × 5 3 .
Converting to Radical Form We can rewrite 2 4 3 in radical form: 2 4 3 = 4 2 3 = 4 8 .
Final Simplification Therefore, the expression becomes 5 3 × 4 8 = 125 4 8 . The value that remains under the radical is 8.
Examples
Understanding radical forms is useful in various fields, such as engineering and physics, where calculations involving roots are common. For instance, when calculating the period of a pendulum, the formula involves a square root. Simplifying radicals helps in obtaining more accurate and manageable results. Also, in geometry, when dealing with areas and volumes of objects, simplifying radicals can make calculations easier and more intuitive. This skill is also essential in computer graphics for rendering images and creating realistic simulations.
After simplifying 125 0 4 3 , the value that remains under the radical is 8. Therefore, the answer is option D, which is 8.
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