Simplify. [tex]$\sqrt{50}+3 \sqrt{18}$[/tex]
Answer (1)
Simplify 50 to 5 2 .
Simplify 18 to 3 2 .
Substitute the simplified expressions into the original expression: 5 2 + 3 ( 3 2 ) .
Combine like terms to get the final answer: 14 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 50 + 3 18 . To do this, we need to simplify the square roots and combine like terms.
Simplifying 50 First, let's simplify 50 . We look for the largest perfect square that divides 50. Since 50 = 25 ⋅ 2 , we can write 50 = 25 ⋅ 2 . Using the property a ⋅ b = a ⋅ b , we have 25 ⋅ 2 = 25 ⋅ 2 = 5 2 .
Simplifying 18 Next, let's simplify 18 . We look for the largest perfect square that divides 18. Since 18 = 9 ⋅ 2 , we can write 18 = 9 ⋅ 2 . Using the property a ⋅ b = a ⋅ b , we have 9 ⋅ 2 = 9 ⋅ 2 = 3 2 .
Substituting Back Now, we substitute the simplified expressions back into the original expression: 50 + 3 18 = 5 2 + 3 ( 3 2 ) = 5 2 + 9 2 .
Combining Like Terms Finally, we combine like terms: 5 2 + 9 2 = ( 5 + 9 ) 2 = 14 2 .
Examples
Square roots are used in many areas of math and science. For example, when calculating the distance between two points in a coordinate plane, you use the distance formula, which involves square roots. If you have two points (1, 2) and (4, 6), the distance between them is ( 4 − 1 ) 2 + ( 6 − 2 ) 2 = 3 2 + 4 2 = 9 + 16 = 25 = 5 . Simplifying square roots helps in finding exact distances and understanding geometric relationships.
