Let’s denote the two numbers as \(x\) and \(y\). We are given that:

1. The sum of the two numbers is 42:
\[x + y = 42\]

2. Their product is 437:
\[xy = 437\]

We are asked to find their difference, which is \(|x – y|\).

To find \(x\) and \(y\), we can use the given sum and product. These can be considered as the sum and product of the roots of a quadratic equation, which can be written as:

\[x^2 – (x + y)x + xy = 0\]

Substituting the given values:

\[x^2 – 42x + 437 = 0\]

Now, we solve this quadratic equation for \(x\). The solutions to the equation will give us the values of \(x\) and \(y\). Instead of directly solving the quadratic equation, let’s use another approach to find the difference, as solving it might get a bit complex without a calculator.

The difference squared can be found using the formula derived from the sum and product of roots:
\[(x – y)^2 = (x + y)^2 – 4xy\]

Given \(x + y = 42\) and \(xy = 437\), we substitute these values into the equation:
\[(x – y)^2 = 42^2 – 4 \cdot 437\]

Calculating the right side:
\[(x – y)^2 = 1764 – 1748\]

\[(x – y)^2 = 16\]

Taking the square root of both sides:
\[|x – y| = 4\]

Therefore, the difference between the two numbers is \(\boldsymbol{4}\).

Given the equation:

\[
5 \sqrt{5} \times 5^3 \div 5^{-3 / 2} = 5^{(a+2)}
\]

We start by expressing all terms as powers of 5:

\[
5^1 \times 5^{\frac{1}{2}} \times 5^3 \div 5^{-\frac{3}{2}} = 5^{a+2}
\]

When you combine the exponents, you add them:

\[
5^{1 + \frac{1}{2} + 3} \times 5^{\frac{3}{2}} = 5^{a+2}
\]

Since multiplying with the same base allows you to add exponents:

\[
5^{\frac{2}{2} + \frac{1}{2} + \frac{6}{2} + \frac{3}{2}} = 5^{a+2}
\]

Simplify the exponents:

\[
5^{\frac{12}{2}} = 5^{a+2}
\]

Which simplifies further to:

\[
5^6 = 5^{a+2}
\]

Setting the exponents equal to each other gives us:

\[
a+2 = 6
\]

Solving for \(a\):

\[
a = 6 – 2
\]
\[
a = 4
\]

Therefore, the value of \(a\) is \(\boldsymbol{4}\).

Given the operation \(x^{*} y = x^{2} + y^{2} – xy\), let’s calculate the value of \(9^{*} 11\).

Substitute \(x = 9\) and \(y = 11\) into the formula:

\[
9^{*} 11 = 9^{2} + 11^{2} – 9 \times 11
\]

Simplify each term:

\[
9^{*} 11 = 81 + 121 – 99
\]

Add and subtract the terms:

\[
9^{*} 11 = 202 – 99 = 103
\]

Therefore, the value of \(9^{*} 11\) is \(\boldsymbol{103}\).

Given \(a^b = 125\) and knowing \(a\) and \(b\) are positive integers, let’s first identify \(a\) and \(b\).

Finding \(a\) and \(b\)

The number 125 is a perfect cube, specifically \(5^3\). Therefore, \(a = 5\) and \(b = 3\).

Calculating \((a-b)^{a+b-4}\)

Now, substitute \(a = 5\) and \(b = 3\) into the expression:

\[
(a-b)^{a+b-4} = (5-3)^{5+3-4}
\]

This simplifies to:

\[
(2)^{4} = 16
\]

Therefore, the value of \((a-b)^{a+b-4}\) is \(\boldsymbol{16}\).

Identification of the Unique Bus Number

The task is to identify a bus number within the range of 1 to 500 that is not only a perfect square but also remains a perfect square when the digits are viewed upside down. Following the clarified criteria:

Criteria for the Bus Number

• The number must be a perfect square.
• When turned upside down, the number must still represent a perfect square.
• The number must include only the digits 0, 1, 6, 8, and 9, as these are the digits that can represent other numbers or themselves when flipped.

Analysis

We consider the range of perfect squares from \(1^2\) to \(22^2\) as these are the perfect squares within the 1 to 500 range:

\[
1, 4, 9, \ldots, 484\ (22^2)
\]

Among these, the number that meets the specific requirement of being a perfect square that, when turned upside down, also becomes a perfect square, is 169:

– Original Number: \(169 = 13^2\), a perfect square.
– Upside Down: When 169 is flipped upside down, it becomes 961. Notably, 6 and 9 flip, while 1 remains the same, making the new figure, 961, which is \(31^2\), another perfect square.

Conclusion

Therefore, the bus number with the described peculiarity is 169. This number is unique in that it satisfies the condition of being a perfect square and also transforms into another perfect square (961) when viewed upside down.