Find the perimeter of the rectangle below.
Show all work for each step. Work should include explanations in words detailing how you simplify each specific radical. Take your time and be thorough. Include units in final answer.
2√6 and 3√54
To find the perimeter of a rectangle, we need to add up all four sides. Let's label the rectangle with its sides:
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| a |
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|____________________|
b
We know that one side (a) has a length of 2√6 and another side (b) has a length of 3√54. Before we can add up the sides, we need to simplify each radical.
Let's start with 2√6. We want to simplify the radical as much as possible. One way to do this is to factor the number inside the radical:
2√6 = 2√(2·3)
Now, we can simplify by taking out any perfect squares from inside the radical. In this case, we can take out a 2:
2√(2·3) = 2√2·√3 = 2√2√3
So, 2√6 simplifies to 2√2√3. We'll use this simplified value in our perimeter calculation.
Now, let's simplify 3√54. Like before, we'll factor the number inside the radical:
3√54 = 3√(2·3·3·3)
Next, we'll take out any perfect squares from inside the radical. We can take out a 9 (which is 3·3), leaving us with:
3√(2·3·9) = 3√2·3√9 = 3√2·3·3
Now, we have simplified 3√54 to 3√2·3·3. We'll use this simplified value in our perimeter calculation.
To calculate the perimeter, we add up all four sides:
Perimeter = 2a + 2b
Substituting in our values for a and b, and simplifying where we can:
Perimeter = 2(2√2√3) + 2(3√2·3·3)
Perimeter = 4√2√3 + 6√2·3·3
Perimeter = 4√2√3 + 18√2
We can't simplify this any further, so we'll keep it in this form. To include units, we'll add them to our final answer:
Perimeter = 4√2√3 + 18√2 units
This is the final answer for the perimeter of the rectangle.