Simplify the radical expression.
Write your answer in the simplest form.
$2 \sqrt{48}+\sqrt{200}-\sqrt{75}-4 \sqrt{32}$
Answer (1)
Simplify 48 to 4 3 .
Simplify 200 to 10 2 .
Simplify 75 to 5 3 .
Simplify 32 to 4 2 .
Substitute and combine like terms: 2 ( 4 3 ) + 10 2 − 5 3 − 4 ( 4 2 ) = 3 3 − 6 2 .
The simplified expression is 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 and then combine like terms.
Simplifying 48 First, let's 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 .
Simplifying 200 Next, let's 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 75 Now, let's 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 32 Finally, let's 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 .
Substituting back into the expression Now we substitute these simplified radicals back into the original expression:
2 48 + 200 − 75 − 4 32 = 2 ( 4 3 ) + 10 2 − 5 3 − 4 ( 4 2 )
= 8 3 + 10 2 − 5 3 − 16 2
Combining like terms Now we combine like terms. We have terms with 3 and terms with 2 .
( 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
Radical expressions are useful in various fields, such as engineering and physics, for calculating lengths, areas, and volumes. For instance, when calculating the diagonal of a square with side length a , the diagonal is a 2 . Simplifying radical expressions helps in obtaining more manageable and understandable forms for these calculations. Imagine you are designing a square garden with an area of 75 square meters. To determine the length of each side, you would need to find 75 , which simplifies to 5 3 meters. This simplified form is easier to work with when planning the layout and fencing for the garden.
