Simplify. [tex]\sqrt{27}+2 \sqrt{12}[/tex]
Answer (1)
Simplify 27 as 3 3 .
Simplify 2 12 as 4 3 .
Combine the simplified terms: 3 3 + 4 3 = 7 3 .
The simplified expression is 7 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 27 + 2 12 . This involves simplifying the square roots and combining like terms.
Simplifying 27 First, we simplify 27 . We look for the largest perfect square that divides 27. Since 27 = 9 × 3 , we have 27 = 9 × 3 = 9 × 3 = 3 3 .
Simplifying 2 12 Next, we simplify 2 12 . We look for the largest perfect square that divides 12. Since 12 = 4 × 3 , we have 2 12 = 2 4 × 3 = 2 ( 4 × 3 ) = 2 ( 2 3 ) = 4 3 .
Combining Like Terms Now, we combine the simplified terms: 27 + 2 12 = 3 3 + 4 3 . Since both terms have a common factor of 3 , we can factor it out: 3 3 + 4 3 = ( 3 + 4 ) 3 = 7 3 .
Final Answer Therefore, the simplified expression is 7 3 .
Examples
Square roots appear in many areas of math and physics. For example, the distance between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) in a plane is given by the distance formula ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . Simplifying expressions with square roots can help in calculating distances or other quantities in geometry and physics.
