Simplify: [tex]3 \sqrt{28}+4 \sqrt{7}+2 \sqrt{32}[/tex]
A) [tex]14 \sqrt{7}+5 \sqrt{2}[/tex]
B) [tex]10 \sqrt{7}+8 \sqrt{2}[/tex]
C) [tex]10 \sqrt{7}+5 \sqrt{2}[/tex]
D) [tex]10 \sqrt{7}-\sqrt{2}[/tex]
Answer (2)
Simplify 28 to 2 7 .
Simplify 32 to 4 2 .
Substitute the simplified forms into the original expression: 3 ( 2 7 ) + 4 7 + 2 ( 4 2 ) .
Combine like terms to get the final answer: 10 7 + 8 2 .
The final answer is 10 7 + 8 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 3 28 + 4 7 + 2 32 . This involves simplifying the square roots and combining like terms.
Simplifying 28 First, we simplify 28 . We can write 28 as 4 × 7 , so 28 = 4 × 7 = 4 × 7 = 2 7 .
Simplifying 32 Next, we simplify 32 . We can write 32 as 16 × 2 , so 32 = 16 × 2 = 16 × 2 = 4 2 .
Substituting Back into the Expression Now we substitute these simplified square roots back into the original expression: 3 28 + 4 7 + 2 32 = 3 ( 2 7 ) + 4 7 + 2 ( 4 2 )
Simplifying the Terms Next, we simplify the expression: 3 ( 2 7 ) + 4 7 + 2 ( 4 2 ) = 6 7 + 4 7 + 8 2
Combining Like Terms Now we combine like terms. We have two terms with 7 and one term with 2 . ( 6 + 4 ) 7 + 8 2 = 10 7 + 8 2
Final Answer The simplified expression is 10 7 + 8 2 . Comparing this to the answer choices, we see that it matches option B.
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 square roots that need simplification. Understanding how to simplify these expressions allows for more accurate and efficient calculations in real-world applications.
The expression 3 28 + 4 7 + 2 32 simplifies to 10 7 + 8 2 , which corresponds to option B. The key steps included simplifying the square roots and combining like terms. Thus, the correct answer is option B.
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