Final answer:
The side length of the smaller square is 7 cm, and the larger square is 14 cm. The area of the larger square is exactly 42 cm² more than the smaller square, with dimensions being twice as long.
Step-by-step explanation:
To determine the length of the sides of each square given that the area of the larger square is 42 cm² more than the area of the smaller square, we can first let the side length of the smaller square be x centimeters. Therefore, the area of the smaller square is x² cm².
Since the area of the larger square is 42 cm² more than that of the smaller square, the larger square's area is x² + 42 cm². Now, if the dimensions of the larger square are twice that of the first square, the side length of the larger square is 2x centimeters, leading to an area of (2x)² which is equal to 4x² cm².
We can set up the equation x² + 42 = 4x² to find x. Solving for x yields x = 7 centimeters, exactly. Thus, the smaller square has sides of 7 cm, and the larger square has sides of 14 cm (7 cm * 2).
An approximate value is not needed in this exact arithmetic problem. If we wanted to approximate, it would simply be rounded to the nearest tenth, which for this exact answer is unnecessary since our results are whole numbers.