A prime number is an integer greater than 1 that has no factors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, and 11.

To simplify this fraction, first find the greatest common factor between them. The factors of 24 are [1, 2, 3, 4, 6, 8, 12, 24] and the factors of 76 are [1, 2, 4, 19, 38, 76]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{24}{76} \) = \( \frac{\frac{24}{4}}{\frac{76}{4}} \) = \( \frac{6}{19} \)

To subtract these radicals together their radicands must be the same:

9\( \sqrt{8} \) - 2\( \sqrt{2} \)
9\( \sqrt{4 \times 2} \) - 2\( \sqrt{2} \)
9\( \sqrt{2^2 \times 2} \) - 2\( \sqrt{2} \)
(9)(2)\( \sqrt{2} \) - 2\( \sqrt{2} \)
18\( \sqrt{2} \) - 2\( \sqrt{2} \)

Now that the radicands are identical, you can subtract them:

The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 75% the radius (and, consequently, the total area) increases by \( \frac{75\text{%}}{2} \) = 37\(\frac{1}{2}\)%

If 77% of the town's 20,000 voters cast ballots the number of votes cast is:

(\( \frac{77}{100} \)) x 20,000 = \( \frac{1,540,000}{100} \) = 15,400

The mayor got 57% of the votes cast which is:

(\( \frac{57}{100} \)) x 15,400 = \( \frac{877,800}{100} \) = 8,778 votes.