Simplify: $19 \sqrt{6}+25 \sqrt{7}+3 \sqrt{6}+6 \sqrt{7}$
Answer (2)
Combine like terms by adding the coefficients of the same square roots.
Add the coefficients of 6 : 19 + 3 = 22 , resulting in 22 6 .
Add the coefficients of 7 : 25 + 6 = 31 , resulting in 31 7 .
Combine the simplified terms to get the final answer: 22 6 + 31 7
Explanation
Understanding the Expression We are asked to simplify the expression $19
\sqrt{6}+25 \sqrt{7}+3 \sqrt{6}+6 \sqrt{7}$. This involves combining like terms, which are terms with the same square root.
Grouping Terms with 6 First, let's group the terms with 6 :
19 6 + 3 6
Adding Coefficients of 6 Now, we add the coefficients of 6 :
19 + 3 = 22 . So, 19 6 + 3 6 = 22 6
Grouping Terms with 7 Next, let's group the terms with 7 :
25 7 + 6 7
Adding Coefficients of 7 Now, we add the coefficients of 7 :
25 + 6 = 31 . So, 25 7 + 6 7 = 31 7
Combining Simplified Terms Finally, we combine the simplified terms:
22 6 + 31 7
Final Result Therefore, the simplified expression is 22 6 + 31 7 .
Examples
Imagine you are calculating the total length of wooden planks needed for a project. Some planks are $\sqrt{6} f ee tl o n g an d o t h ers a re \sqrt{7} f ee tl o n g . I f yo u ha v e 19 pl ank so f l e n g t h \sqrt{6} f ee t , 3 pl ank so f l e n g t h \sqrt{6} f ee t , 25 pl ank so f l e n g t h \sqrt{7} f ee t , an d 6 pl ank so f l e n g t h \sqrt{7}$ feet, you can use the simplification above to quickly determine the total combined length of each type of plank. This helps in estimating the total material required for the project.
To simplify the expression 19 6 + 25 7 + 3 6 + 6 7 , we combined like terms to get 22 6 + 31 7 . This involves adding together the coefficients for terms with the same square root. The final simplified result is 22 6 + 31 7 .
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