The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

To simplify this fraction, first find the greatest common factor between them. The factors of 32 are [1, 2, 4, 8, 16, 32] and the factors of 72 are [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72]. They share 4 factors [1, 2, 4, 8] making 8 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{32}{72} \) = \( \frac{\frac{32}{8}}{\frac{72}{8}} \) = \( \frac{4}{9} \)

You have 36 out of the total of 450 raffle tickets sold so you have a (\( \frac{36}{450} \)) x 100 = \( \frac{36 \times 100}{450} \) = \( \frac{3600}{450} \) = 8% chance to win the raffle.

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so add the coefficients and retain the base and exponent:

-5z⁷ + 4z⁷
(-5 + 4)z⁷
-z⁷

To multiply terms with exponents, the base of both exponents must be the same. In this case they are so multiply the coefficients and add the exponents:

-y⁵ x 5y³
(-1 x 5)y^{(5 + 3)}
-5y⁸