What is the simplified form of the following expression?
$5 \sqrt{8}-\sqrt{18}-2 \sqrt{2}$
Answer (1)
Simplify 8 to 2 2 .
Simplify 18 to 3 2 .
Substitute the simplified radicals into the expression: 5 ( 2 2 ) − 3 2 − 2 2 .
Combine like terms to get the final answer: 5 2 .
Explanation
Analyze the problem Let's simplify the expression step-by-step. Our goal is to combine the terms by expressing each square root in terms of 2 .
Simplify 8 First, we simplify 8 . We can rewrite 8 as 4 ⋅ 2 . Therefore, 8 = 4 ⋅ 2 = 4 ⋅ 2 = 2 2 .
Simplify 18 Next, we simplify 18 . We can rewrite 18 as 9 ⋅ 2 . Therefore, 18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2 .
Substitute back into the expression Now, we substitute the simplified radicals back into the original expression: 5 8 − 18 − 2 2 = 5 ( 2 2 ) − 3 2 − 2 2 .
Simplify the expression We simplify further: 5 ( 2 2 ) − 3 2 − 2 2 = 10 2 − 3 2 − 2 2 .
Combine like terms Now, we combine the like terms: 10 2 − 3 2 − 2 2 = ( 10 − 3 − 2 ) 2 = ( 7 − 2 ) 2 = 5 2 .
Final Answer Therefore, the simplified form of the expression is 5 2 .
Examples
Square roots appear in various fields, such as physics and engineering. For example, when calculating the length of the diagonal of a square with side length a , we use the Pythagorean theorem, which gives us a diagonal of length a 2 + a 2 = 2 a 2 = a 2 . Simplifying radical expressions helps in obtaining more manageable and understandable forms of equations and solutions in these practical applications.
