Simplify: [tex]7 \sqrt{5}-4 \sqrt{7}+9 \sqrt{5}+5 \sqrt{7}[/tex]
A) [tex]16 \sqrt{5}+\sqrt{7}[/tex]
B) [tex]17 \sqrt{12}[/tex]
C) [tex]16 \sqrt{10}+\sqrt{14}[/tex]
D) [tex]3 \sqrt{5}+13 \sqrt{7}[/tex]
Answer (1)
Combine like terms: Group the terms with 5 and 7 separately.
Simplify 5 terms: 7 5 + 9 5 = 16 5 .
Simplify 7 terms: − 4 7 + 5 7 = 7 .
Combine the simplified terms: The final simplified expression is 16 5 + 7 .
Explanation
Understanding the Problem We are asked to simplify the expression 7 5 − 4 7 + 9 5 + 5 7 . This involves combining like terms, specifically the terms with 5 and the terms with 7 .
Combining Like Terms First, let's combine the terms with 5 :
7 5 + 9 5 = ( 7 + 9 ) 5 = 16 5
Next, let's combine the terms with 7 :
− 4 7 + 5 7 = ( − 4 + 5 ) 7 = 1 7 = 7
Now, we add the simplified terms together:
16 5 + 7
Final Answer The simplified expression is 16 5 + 7 . Comparing this to the given options, we see that it matches option A.
Conclusion Therefore, the simplified expression is 16 5 + 7 .
Examples
Radical expressions are useful in various fields, such as engineering and physics, for calculating lengths, areas, and volumes. For example, when calculating the diagonal of a square with side length s , the diagonal is s 2 . Simplifying radical expressions helps in obtaining more manageable and understandable results in practical applications.
