Select the correct answer.
What is this expression in simplified form?
[tex]$3 \sqrt{3} \cdot 6 \sqrt{6}$[/tex]
A. [tex]$18 \sqrt{2}$[/tex]
B. [tex]$54 \sqrt{2}$[/tex]
C. [tex]$18 \sqrt{3}$[/tex]
D. 54
Answer (1)
Multiply the coefficients: 3 ⋅ 6 = 18 .
Multiply the terms inside the square roots: 3 ⋅ 6 = 18 .
Simplify the square root: 18 = 3 2 .
Multiply the results: 18 ⋅ 3 2 = 54 2 . The answer is 54 2 .
Explanation
Understanding the Expression Let's simplify the expression step-by-step to make sure we understand each part. We're starting with 3 3 ⋅ 6 6 .
Multiplying Coefficients First, we can multiply the numbers outside the square roots. So, we multiply 3 and 6:
3 ⋅ 6 = 18
Multiplying Radicands Next, we multiply the terms inside the square roots: 3 ⋅ 6 = 3 ⋅ 6 = 18
Simplifying the Square Root Now, we simplify the square root of 18. We can rewrite 18 as 9 ⋅ 2 . So, 18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2
Combining the Results Finally, we multiply the result from step 2 with the simplified square root from step 4: 18 ⋅ 3 2 = 54 2
Final Answer So, the simplified form of the expression is 54 2 . Therefore, the correct answer is B.
Examples
Imagine you're calculating the area of a rectangle where the sides involve square roots. For instance, one side might be 3 3 meters and the other 6 6 meters. To find the area, you multiply these two lengths together. Simplifying such expressions is not just a math exercise; it's a practical skill in geometry and physics, helping you determine areas, volumes, and other quantities in a simplified and understandable form. This skill is also crucial in fields like engineering and architecture, where precise calculations are essential for design and construction.
