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A square has an area of $4320\,\text{m}^2$ . Determine the perimeter of the square. Write the answer as a radical in simplest form.

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Roots of integers

Full solution

Q. A square has an area of

4320\,\text{m}^2

. Determine the perimeter of the square. Write the answer as a radical in simplest form.

Find Side Length: To find the perimeter, first find the length of one side of the square by taking the square root of the area. $\newline$ $\sqrt{4320} \, \text{m}^2 = ?$
Simplify Square Root: Simplify the square root by factoring out perfect squares. $\newline$ $4320 = 2^5 \times 3^3 \times 5$ $\newline$ $\sqrt{4320} = \sqrt{(2^5 \times 3^3 \times 5)} = \sqrt{(2^4 \times 2 \times 3^2 \times 3 \times 5)} = 2^2 \times 3 \times \sqrt{(2 \times 3 \times 5)}$ $\newline$ $\sqrt{4320} = 4 \times 3 \times \sqrt{(30)} = 12\sqrt{(30)}$ m
Calculate Perimeter: The perimeter of a square is $4$ times the length of one side. $\newline$ Perimeter = $4 \times \text{side}$ $\newline$ Perimeter = $4 \times 12\sqrt{30}$ m
Final Perimeter Calculation: Calculate the perimeter. $\newline$ Perimeter = $48\sqrt{30}$ m

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