The binary representation of the decimal number 8 is 1000.

The decimal number 10 is represented as 1010 in binary.

The quadratic x^2 - 5x + 6 factors to (x - 2)(x - 3).

The expression factors to (x - 2)(x - 3) since -2 and -3 are the roots.

log_5(1) = 0 because any number raised to the power of 0 equals 1.

The correct order is to first solve the parentheses (4 - 1), then multiply by 3, and finally add 2.

Binary is the simplest (base-2), followed by Decimal (base-10), and then Hexadecimal (base-16), making the correct order Binary, Decimal, Hexadecimal.

All the listed properties are correct and fundamental to logarithmic functions.

15 divided by 4 gives a remainder of 3, hence 15 mod 4 = 3.

To convert 10 to base-3, divide by 3: 10 ÷ 3 = 3 remainder 1, 3 ÷ 3 = 1 remainder 0, 1 ÷ 3 = 0 remainder 1. Reading the remainders from bottom to top gives 110.

In base 8, the decimal number 15 is represented as '17'.

The decimal number 15 is represented as '1111' in binary.

In hexadecimal, the decimal number 15 is represented as F.

15 in decimal is represented as '33' in base-4 (3*4^1 + 3*4^0 = 12 + 3 = 15).

25 in decimal is represented as '31' in base-8 (3*8^1 + 1*8^0 = 24 + 1 = 25).

Using the quotient rule for exponents, (x^2y^3)/(xy^2) simplifies to x^(2-1)y^(3-2) = xy.

Using the property of exponents for division, we subtract the exponents: x^(3-2) * y^(2-1) = x^1 * y^1.

Using the power of a product property, (a*b)^n = a^n * b^n, we get (x^3)^2 * (y^2)^2 = x^6 * y^4.

Using the power of a product property, (x^3y^2)^2 = x^(3*2)y^(2*2) = x^6y^4.

log_10(1000) = 3 and log_10(0.01) = -2, thus 3 + (-2) = 1.

Using the property of logarithms, log_10(1000) = 3 and log_10(10) = 1. Therefore, log_10(1000) - log_10(10) = 3 - 1 = 2.

log_10(1000) = log_10(10^3) = 3.

Using the property of logarithms, log_2(8) + log_2(4) = log_2(8*4) = log_2(32) = log_2(16).

log_5(25) = 2 and log_5(5) = 1, thus log_5(25) - log_5(5) = 2 - 1 = 1.

Using the power rule of logarithms, log_a(b^2) simplifies to 2 log_a(b).

The property of logarithms states that log_a(b^c) = c * log_a(b).

Completing the square for the equation y = x² - 4x + 3 results in y = (x - 2)² - 1.

The GCD of 48 and 180 is 12, as it is the largest number that divides both without a remainder.

The GCF of 48 and 180 is 12.

The LCM of 9 and 12 is 36, as it is the smallest number that both can divide evenly.