Question

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Answer 1

`\sf 2(3√(8)+4√(20)+2√(24))`

`\sf 2(3√(8))+2(4√(20)+2√(24))`

`\sf 12 √(2)+16 √(5)+8 √(6)`

`\sf 6 \cdot 8^{(1)/(2)}+8 \cdot 20^{(1)/(2)}+4 \cdot 24^{(1)/(2)}`

`\sf 12 \cdot 2^{(1)/(2)}+16 \cdot 5^{(1)/(2)}+8 \cdot 6^{(1)/(2)}`

Explanation:

Perimeter of a rectangle

Perimeter = 2(width + length)

Given information:

`\textsf{width} = 3√(8)`

`\textsf{length} = 4√(20)+2√(24)`

Equivalent Expression 1

Substitute the given information into the formula:

`\sf Perimeter = 2(3√(8)+4√(20)+2√(24))`

Equivalent Expression 2

Using the distributive property law, this can also be written as:

`\sf Perimeter = 2(3√(8))+2(4√(20)+2√(24))`

Equivalent Expression 3

Distribute the parentheses and simplify the radicals:

`\begin{aligned}\sf Perimeter & = \sf 2(3√(8)+4√(20)+2√(24))\\& = \sf 6√(8)+8√(20)+4√(24)\\& = \sf 6√(4 \cdot 2)+8√(4 \cdot 5)+4√(4 \cdot 6)\\& = \sf 6√(4)√(2)+8√(4)√(5)+4√(4)√(6)\\& = \sf 6 \cdot 2 √(2)+8 \cdot 2 √(5)+4 \cdot 2 √(6)\\& = \sf 12 √(2)+16 √(5)+8 √(6)\end{aligned}`

Equivalent Expression 4

Distribute the parentheses and rewrite the square roots as
`\sf √(a)=a^{(1)/(2)}` :

`\begin{aligned}\sf Perimeter & = \sf 2(3√(8)+4√(20)+2√(24))\\& = \sf 6√(8)+8√(20)+4√(24)\\ & = \sf 6 \cdot 8^{(1)/(2)}+8 \cdot 20^{(1)/(2)}+4 \cdot 24^{(1)/(2)}\end{aligned}`

Equivalent Expression 5

Rewrite the square roots as
`\sf √(a)=a^{(1)/(2)}` :

`\sf Perimeter=12 \cdot 2^{(1)/(2)}+16 \cdot 5^{(1)/(2)}+8 \cdot 6^{(1)/(2)}`