$7 \sqrt{3}-2 \sqrt{2}+3 \sqrt{2}+5 \sqrt{3}$
Answer (1)
Group like terms: ( 7 3 + 5 3 ) + ( − 2 2 + 3 2 )
Combine coefficients of like terms: 12 3 + 2
The simplified expression is: 12 3 + 2
Explanation
Understanding the Expression We are asked to simplify the expression 7 3 − 2 2 + 3 2 + 5 3 . This involves combining like terms, which are terms that have the same radical part.
Grouping Like Terms First, let's group the like terms together. We have terms with 3 and terms with 2 . So we can rewrite the expression as: ( 7 3 + 5 3 ) + ( − 2 2 + 3 2 )
Combining Coefficients Now, we can combine the coefficients of the like terms. For the terms with 3 , we have 7 3 + 5 3 = ( 7 + 5 ) 3 = 12 3 .
For the terms with 2 , we have − 2 2 + 3 2 = ( − 2 + 3 ) 2 = 1 2 = 2 .
Final Simplified Expression So, the simplified expression is 12 3 + 2 .
Examples
Radicals are useful in many areas, including engineering and physics. For example, when calculating the length of the diagonal of a square with side length s , we use the Pythagorean theorem to find that the diagonal has length s 2 . If we have multiple squares and want to find the total length of their diagonals, we can use the same simplification techniques with radicals as shown above.
