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Here are the explanations for √3, √5, and √7:

√3:
*Geometric Method:*
Imagine an equilateral triangle with side length 2 units. Draw an altitude from one vertex to the opposite side, creating two 30-60-90 right triangles.

Using the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the legs, and c is the hypotenuse.

1^2 + (√3)^2 = 2^2
1 + 3 = 4
√3 = √3

Now, imagine a hexagon with side length 1 unit. The diagonal of this hexagon is equal to √3 units.

Using a compass and straightedge, we can construct a sequence of triangles and hexagons to approximate the value of √3.

*Decimal Expansion:*
We can express √3 as a decimal using the following method:

√3 = 1 + 0.7 + 0.04 + 0.002 + ... (using the Babylonian method or other numerical methods)

This decimal expansion converges to approximately 1.7321.

√5:
*Geometric Method:*
Imagine a golden rectangle with length 2 units and width 1 unit. Draw a diagonal, creating two similar triangles.

Using the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the legs, and c is the hypotenuse.

1^2 + 2^2 = (√5)^2
1 + 4 = 5
√5 = √5

Now, imagine a decagon with side length 1 unit. The diagonal of this decagon is equal to √5 units.

Using a compass and straightedge, we can construct a sequence of triangles and decagons to approximate the value of √5.

*Decimal Expansion:*
We can express √5 as a decimal using the following method:

√5 = 2 + 0.2 + 0.01 + 0.0005 + ... (using the Babylonian method or other numerical methods)

This decimal expansion converges to approximately 2.2361.

√7:
*Geometric Method:*
Imagine a heptagon with side length 1 unit. Draw a diagonal, creating two similar triangles.

Using the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the legs, and c is the hypotenuse.

1^2 + (√7)^2 = (√14)^2
1 + 7 = 14
√7 = √7

Now, imagine a 14-gon with side length 1 unit. The diagonal of this 14-gon is equal to √7 units.

Using a compass and straightedge, we can construct a sequence of triangles and 14-gons to approximate the value of √7.

*Decimal Expansion:*
We can express √7 as a decimal using the following method:

√7 = 2 + 0.6 + 0.03 + 0.0015 + ... (using the Babylonian method or other numerical methods)

This decimal expansion converges to approximately 2.6458.

Here are the approximate values:

- √3 ≈ 1.7321
- √5 ≈ 2.2361
- √7 ≈ 2.6458

To understand how √2 equals approximately 1.4142, we can use various mathematical techniques. Here's a broad explanation:

1. Geometric Method:
Imagine a right-angled triangle with both legs equal to 1 unit. Using the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the legs, and c is the hypotenuse.

1^2 + 1^2 = c^2
2 = c^2
c = √2

Now, imagine a square with a side length of 1 unit. The diagonal of this square is equal to √2 units.

Using a compass and straightedge, we can construct a sequence of triangles and squares to approximate the value of √2.

2. Decimal Expansion:
We can express √2 as a decimal using the following method:

√2 = 1 + 0.4 + 0.02 + 0.0004 + ... (using the Babylonian method or other numerical methods)

This decimal expansion converges to approximately 1.4142.

3. Continued Fractions:
We can express √2 as a continued fraction:

√2 = 1 + 1/(2 + 1/(2 + 1/(2 + ...)))

This continued fraction converges to approximately 1.4142.

4. Babylonian Method:
This ancient method involves making an initial guess, then iteratively refining that guess using the formula:

x(n+1) = (x(n) + 2/x(n)) / 2

Starting with an initial guess of 1, we can iterate this formula to converge to approximately 1.4142.

These mathematical techniques demonstrate how √2 equals approximately 1.4142.

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