Find the area of the figure 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 the final answer. Use radicals in your work and not decimals!
image: Rectangle with a base of (7√3+ 5√6)
Height (-2√3+ 3√6)
I wanted to check to see if I was right I used the foil method
(7√3+ 5√6) x (-2√3+ 3√6)
-14√9 + -21√18 + -10√18 + 15√36
-14x3 + 11√18 + 15x6
-42+11√18+90
48 + 11√18in
so far, so good, but note that √18 = 3√2
To find the area of the rectangle, we need to multiply the length (base) and the width (height) of the rectangle.
Given:
Length (base) = 7√3 + 5√6
Width (height) = -2√3 + 3√6
Step 1: Identify the lengths of the sides
The length of the rectangle is 7√3 + 5√6, and the width is -2√3 + 3√6.
Step 2: Multiply the lengths
To find the area, we need to multiply the length and width. Using the distributive property, we multiply each term of the length by each term of the width and then combine like terms.
Area = (7√3 + 5√6) * (-2√3 + 3√6)
Step 3: Simplify the multiplication
To multiply the terms inside the parentheses, we can use the FOIL method (First, Outer, Inner, Last).
Area = [7√3 * -2√3] + [7√3 * 3√6] + [5√6 * -2√3] + [5√6 * 3√6]
Step 4: Simplify each multiplication
Now we can calculate each term separately.
Area = -2 * 7 * (√3 * √3) + 7 * 3 * (√3 * √6) - 2 * 5 * (√6 * √3) + 5 * 3 * (√6 * √6)
Step 5: Simplify the square roots
√3 * √3 = 3, and √6 * √6 = 6.
Area = -2 * 7 * 3 + 7 * 3 * (√3 * √6) - 2 * 5 * (√6 * √3) + 5 * 3 * 6
Step 6: Simplify the multiplication
Now we can calculate the multiplications without square roots.
Area = -42 + 42(√3 * √6) - 10(√6 * √3) + 90
Step 7: Simplify the multiplications with square roots
√3 * √6 = √(3*6) = √18 = √(9*2) = √9 * √2 = 3√2
√6 * √3 = √(6*3) = √18 = 3√2
Area = -42 + 42(3√2) - 10(3√2) + 90
Step 8: Combine like terms
Combine the terms without square roots and the terms with square roots separately.
Area = -42 + 126√2 - 30√2 + 90
Step 9: Combine all terms
Combine all the terms together.
Area = 48√2 + 48
Step 10: Simplify the final answer
The area of the given rectangle is 48√2 + 48 square units.