Simplify each expression. $-\(\frac{1}{4}\)(20m + 8y - 32z)$

Verified step by step guidance

1. Start by distributing the factor \(-\frac{1}{4}\) to each term inside the parentheses: \(-\frac{1}{4} \times 20m\), \(-\frac{1}{4} \times 8y\), and \(-\frac{1}{4} \times (-32z)\).
2. Multiply each term separately: For \(-\frac{1}{4} \times 20m\), multiply the coefficients \(-\frac{1}{4}\) and \(20\) and keep the variable \(m\); do the same for the other terms.
3. Remember that multiplying two negative numbers results in a positive number, so pay attention to the sign when multiplying \(-\frac{1}{4}\) and \(-32z\).
4. After multiplying, write down the simplified terms: the product of \(-\frac{1}{4}\) and \$20m\(, the product of \(-\frac{1}{4}\) and \)8y\(, and the product of \(-\frac{1}{4}\) and \)-32z$.
5. Combine all the simplified terms to write the final simplified expression.

Key Concepts

Distributive Property

The distributive property allows you to multiply a single term outside the parentheses by each term inside the parentheses. For example, a(b + c) = ab + ac. This property is essential for simplifying expressions like -1/4(20m + 8y - 32z) by distributing -1/4 to each term.

Multiplying Fractions and Variables

When multiplying a fraction by a term with variables, multiply the numerator by the term and keep the denominator. Variables remain attached to their coefficients during multiplication. For instance, (-1/4) × 20m equals (-1 × 20m)/4 = -5m.

Combining Like Terms

After distributing, expressions may have terms that can be combined if they have the same variable and exponent. Combining like terms simplifies the expression further. In this problem, each term is distinct, so combining like terms may not apply, but understanding this concept is important for simplification.