What is the simplified form of the following expression? [tex]$2 \sqrt{27}+\sqrt{12}-3 \sqrt{3}-2 \sqrt{12}$[/tex]
Answer (2)
The simplified form of the expression 2 27 + 12 − 3 3 − 2 12 is 3 . This is achieved by simplifying each square root and then combining like terms. Therefore, the final answer is 3 .
;
Simplify 27 to 3 3 .
Simplify 12 to 2 3 .
Substitute the simplified radicals into the original expression: 2 ( 3 3 ) + 2 3 − 3 3 − 2 ( 2 3 ) .
Combine like terms to get 3 .
3
Explanation
Understanding the Problem We are asked to simplify the expression 2 27 + 12 − 3 3 − 2 12 . To do this, we need to simplify the radicals and combine like terms.
Simplifying the Radicals First, we simplify 27 . We can write 27 as 9 × 3 , so 27 = 9 × 3 = 9 × 3 = 3 3 .
Simplifying the Radicals Next, we simplify 12 . We can write 12 as 4 × 3 , so 12 = 4 × 3 = 4 × 3 = 2 3 .
Substituting the Simplified Radicals Now we substitute these simplified radicals back into the original expression: 2 27 + 12 − 3 3 − 2 12 = 2 ( 3 3 ) + ( 2 3 ) − 3 3 − 2 ( 2 3 )
Simplifying the Expression Now we simplify the expression: 2 ( 3 3 ) + 2 3 − 3 3 − 2 ( 2 3 ) = 6 3 + 2 3 − 3 3 − 4 3
Combining Like Terms Now we combine like terms: 6 3 + 2 3 − 3 3 − 4 3 = ( 6 + 2 − 3 − 4 ) 3 = ( 8 − 7 ) 3 = 1 3 = 3
Final Answer Therefore, the simplified form of the expression is 3 .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, when dealing with lengths, areas, or volumes that involve square roots. For example, when calculating the diagonal of a square or the distance between two points, you often encounter radical expressions that need simplification to obtain the most concise and understandable form.
