Simplify: $21 \sqrt{3}+19 \sqrt{5}+8 \sqrt{3}+12 \sqrt{5}$
A. $8 \sqrt{3}+31 \sqrt{5}$
B. $29 \sqrt{3}+31 \sqrt{5}$
C. $29 \sqrt{5}+31 \sqrt{3}$
D. $29 \sqrt{3}-31 \sqrt{5}$
Answer (2)
The simplified expression is 29 3 + 31 5 .
Explanation
Identifying Like Terms Let's simplify the expression by combining like terms. Like terms are terms that have the same radical part. In this case, we have terms with 3 and terms with 5 .
Grouping Like Terms First, let's group the like terms together: ( 21 3 + 8 3 ) + ( 19 5 + 12 5 )
Combining Coefficients Now, we can combine the coefficients of the like terms. This means adding the numbers in front of the radicals: ( 21 + 8 ) 3 + ( 19 + 12 ) 5
Simplifying Coefficients Let's simplify the coefficients by performing the addition: 29 3 + 31 5
Final Answer So, the simplified expression is 29 3 + 31 5 .
Examples
Imagine you are calculating the total length of wooden planks needed for a project. You have planks of length 3 meters and 5 meters. If you have 29 planks of 3 meters and 31 planks of 5 meters, the expression 29 3 + 31 5 represents the total length of wood you need. This kind of simplification is useful in various fields like construction, physics, and engineering where you deal with quantities involving radicals.
After simplifying the expression 21 3 + 19 5 + 8 3 + 12 5 , we find that it simplifies to 29 3 + 31 5 . The correct answer from the given options is B. 29 3 + 31 5 .
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