Rewrite each expression using the distributive property and simplify, if possible. See Example 7. 3 16 32 40—— ( —— y + —— z - —— ) 8 9 27 9

Verified step by step guidance

1. Identify the expression to be distributed: \( \frac{3}{8} \left( \frac{16}{9}y + \frac{32}{27}z - \frac{40}{9} \right) \).
2. Apply the distributive property: distribute \( \frac{3}{8} \) to each term inside the parentheses.
3. Calculate \( \frac{3}{8} \times \frac{16}{9}y \) to simplify the first term.
4. Calculate \( \frac{3}{8} \times \frac{32}{27}z \) to simplify the second term.
5. Calculate \( \frac{3}{8} \times \left(-\frac{40}{9}\right) \) to simplify the third term.

Key Concepts

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing us to multiply a single term by each term within a parenthesis. This property is essential for simplifying expressions, as it helps to eliminate parentheses and combine like terms effectively.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD). This process is crucial in making expressions more manageable and easier to work with, especially when performing operations like addition or subtraction.

Combining Like Terms

Combining like terms is the process of adding or subtracting terms that have the same variable raised to the same power. This step is vital in simplifying algebraic expressions, as it consolidates terms to create a more concise and understandable form of the expression.