What is the following difference?
[tex]$11 \sqrt{45}-4 \sqrt{5}$[/tex]
A. [tex]$95 \sqrt{5}$[/tex]
B. [tex]$29 \sqrt{5}$[/tex]
C. [tex]$14 \sqrt{10}$[/tex]
D. [tex]$7 \sqrt{40}$[/tex]
Answer (2)
Simplify 45 to 3 5 .
Substitute the simplified radical into the expression: 11 ( 3 5 ) − 4 5 .
Simplify the expression to 33 5 − 4 5 .
Combine like terms to get the final answer: 29 5 .
Explanation
Understanding the Problem We are given the expression 11 45 − 4 5 and asked to simplify it and choose the correct answer from the list of options.
Simplifying the Radical First, we need to simplify the radical 45 . We can rewrite 45 as 9 × 5 , so 45 = 9 × 5 = 9 × 5 = 3 5 .
Substituting Back into the Expression Now, substitute this simplified radical back into the original expression: 11 ( 3 5 ) − 4 5 .
Simplifying the Expression Next, we simplify the expression: 33 5 − 4 5 .
Combining Like Terms Now, combine like terms: ( 33 − 4 ) 5 = 29 5 .
Comparing with Options Finally, we compare the simplified expression 29 5 with the given options.
Examples
Radical expressions are useful in many areas, including engineering and physics. For example, when calculating the period of a pendulum, you use a formula that includes a square root. Simplifying radical expressions makes these calculations easier. Also, when dealing with lengths and distances, simplifying radicals can help in obtaining more manageable and understandable results. For instance, if you are building a triangular structure and one side has a length of 11 45 meters and another side has a length of 4 5 meters, finding the difference in their lengths involves simplifying the expression 11 45 − 4 5 , which simplifies to 29 5 meters.
The simplified form of the expression 11 45 − 4 5 is 29 5 . Therefore, the correct answer is option B. This solution involved simplifying 45 to obtain 3 5 and then combining like terms.
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