Question

write each expression in simplest radical form.

1. 18\sqrt{12}+6
2. 4\sqrt{8}+6\sqrt{8}+5\sqrt{12}+2\sqrt{12}
3. the area of a rectangle is (5\sqrt{5})\times(3\sqrt{11}) cm². what is the area of the rectangle in simplest form?
4. a rectangle has length = 2 + \sqrt{3} and width = 3 - \sqrt{5}. find the perimeter and area of the rectangle.

perimeter =
area =

Explanation:

Step1: Simplify $\sqrt{12}$ in 12)

$\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}$, so $18\sqrt{12}+6 = 18\times2\sqrt{3}+6=36\sqrt{3}+6$

Step2: Combine like - terms in 13)

First, $4\sqrt{8}+6\sqrt{8}=(4 + 6)\sqrt{8}=10\sqrt{8}=10\times2\sqrt{2}=20\sqrt{2}$, and $5\sqrt{12}+2\sqrt{12}=(5 + 2)\sqrt{12}=7\sqrt{12}=14\sqrt{3}$

Step3: Multiply radicals in 14)

$(5\sqrt{5})\times(3\sqrt{11})=5\times3\times\sqrt{5\times11}=15\sqrt{55}$

Step4: Calculate perimeter in 15)

Perimeter of rectangle $P = 2(l + w)=2((2+\sqrt{3})+(3-\sqrt{5}))=2(5+\sqrt{3}-\sqrt{5})=10 + 2\sqrt{3}-2\sqrt{5}$

Step5: Calculate area in 15)

Area of rectangle $A=l\times w=(2+\sqrt{3})(3 - \sqrt{5})=2\times3-2\sqrt{5}+3\sqrt{3}-\sqrt{15}=6 - 2\sqrt{5}+3\sqrt{3}-\sqrt{15}$

Answer:

1. $36\sqrt{3}+6$
2. $20\sqrt{2}+7\sqrt{12}=20\sqrt{2}+14\sqrt{3}$
3. $15\sqrt{55}$
4. Perimeter: $10 + 2\sqrt{3}-2\sqrt{5}$; Area: $6 - 2\sqrt{5}+3\sqrt{3}-\sqrt{15}$