Answer: 4 * 2 ^ (1/2).
Let the side of the square be a, then its area is a ^ 2, and the perimeter is 2 * 2 * a = 4 * a. If the area is increased 32 times, then the side of the square will increase. A new square with side b = a + x, where x is the difference by which the side was increased, then the area will be b ^ 2 = 32 * a ^ 2 => b = 4 * a * 2 ^ (1/2), perimeter 4 * b = 4 * 4 * a * 2 ^ (1/2) => when comparing the perimeters of 4 * a and 16 * a * 2 ^ (1/2), it follows that the perimeter has increased by (16 * a * 2 ^ (1 / 2)) / 4 * a = 4 * 2 ^ (1/2).
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