Simplify the radical expression. Write your answer in the simplest form.
[tex]2 \sqrt{48}+\sqrt{200}-\sqrt{75}-4 \sqrt{32}[/tex]
Answer (2)
Simplify each radical term: 48 = 4 3 , 200 = 10 2 , 75 = 5 3 , 32 = 4 2 .
Substitute the simplified radicals into the expression: 2 ( 4 3 ) + 10 2 − 5 3 − 4 ( 4 2 ) .
Combine like terms: ( 8 3 − 5 3 ) + ( 10 2 − 16 2 ) .
Simplify to get the final answer: 3 3 − 6 2 .
Explanation
Understanding the problem We are asked to simplify the expression 2 48 + 200 − 75 − 4 32 . To do this, we need to simplify each radical term individually and then combine like terms.
Simplifying the first term First, we simplify 48 . We look for the largest perfect square that divides 48. Since 48 = 16 × 3 , we have 48 = 16 × 3 = 16 × 3 = 4 3 . Therefore, 2 48 = 2 × 4 3 = 8 3 .
Simplifying the second term Next, we simplify 200 . We look for the largest perfect square that divides 200. Since 200 = 100 × 2 , we have 200 = 100 × 2 = 100 × 2 = 10 2 .
Simplifying the third term Now, we simplify 75 . We look for the largest perfect square that divides 75. Since 75 = 25 × 3 , we have 75 = 25 × 3 = 25 × 3 = 5 3 .
Simplifying the fourth term Finally, we simplify 32 . We look for the largest perfect square that divides 32. Since 32 = 16 × 2 , we have 32 = 16 × 2 = 16 × 2 = 4 2 . Therefore, 4 32 = 4 × 4 2 = 16 2 .
Substituting back into the original expression Now we substitute these simplified expressions back into the original expression: 2 48 + 200 − 75 − 4 32 = 8 3 + 10 2 − 5 3 − 16 2 .
Combining like terms We combine like terms: 8 3 − 5 3 + 10 2 − 16 2 = ( 8 − 5 ) 3 + ( 10 − 16 ) 2 = 3 3 − 6 2 .
Final Answer Therefore, the simplified expression is 3 3 − 6 2 .
Examples
Simplifying radical expressions is useful in various fields, such as engineering and physics, where calculations often involve square roots. For example, when calculating the impedance of an electrical circuit or determining the distance between two points in a coordinate system, simplifying radicals can make the final result easier to interpret and use. Imagine you are designing a bridge and need to calculate the length of a support beam that involves a square root. Simplifying the radical will give you a more manageable number to work with, ensuring accuracy and efficiency in your design.
The expression 2 48 + 200 − 75 − 4 32 simplifies to 3 3 − 6 2 after breaking down each radical and combining like terms. Each radical is simplified separately by factoring out perfect squares. The final result is obtained by combining the simplified terms appropriately.
;
