Select the correct answer. What is this expression in simplified form? [tex]2 \sqrt{2} \cdot 7 \sqrt{18}[/tex] A. [tex]$18 \sqrt{5}$[/tex] B. 84 C. [tex]$84 \sqrt{2}$[/tex] D. [tex]$14 \sqrt{6}$[/tex]
Answer (2)
Simplify 18 to 3 2 .
Substitute the simplified form into the original expression: 2 2 ".7 ( 3 2 ) .
Multiply the constants: ( 2".7".3 ) ( 2 ". 2 ) = 42".2 .
Simplify to get the final answer: 84 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 s q r t 2 ⋅ 7 18 and choose the correct answer from the given options.
Simplifying the Square Root First, we can simplify 18 . Since 18 = 9 ⋅ 2 , we have 18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2 .
Substituting Back Now we substitute this back into the original expression: 2 2 ⋅ 7 18 = 2 2 ⋅ 7 ( 3 2 ) .
Multiplying Constants and Square Roots Next, we rearrange and multiply the constants: ( 2 ⋅ 7 ⋅ 3 ) ( 2 ⋅ 2 ) = ( 42 ) ( 2 ⋅ 2 ) . Since 2 ⋅ 2 = 2 , we have ( 42 ) ( 2 ) = 84 .
Final Answer Therefore, the simplified expression is 84. Looking at the answer choices, we see that option B is 84.
Examples
Understanding how to simplify radical expressions is useful in many areas, such as physics and engineering, where you might need to calculate distances or areas. For example, if you are designing a square garden with an area of 18 m 2 , the length of each side would be 18 meters. Simplifying this to 3 2 meters can make it easier to work with in further calculations or when measuring out the garden. This skill is also crucial in more advanced mathematical contexts, such as calculus and linear algebra, where simplifying expressions is a common task.
The expression 2 2 ⋅ 7 18 simplifies to 84 . Therefore, the correct answer is option B. 84.
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