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. First, rewrite the given expression clearly: \(\frac{3}{8} \left( \frac{16}{9} y + \frac{32}{27} z - \frac{40}{9} \right)\).
2. Apply the distributive property by multiplying \(\frac{3}{8}\) with each term inside the parentheses separately: \(\frac{3}{8} \times \frac{16}{9} y\), \(\frac{3}{8} \times \frac{32}{27} z\), and \(\frac{3}{8} \times \left(-\frac{40}{9}\right)\).
3. Multiply the numerators together and the denominators together for each term: For example, \(\frac{3 \times 16}{8 \times 9} y\), and similarly for the other terms.
4. Simplify each fraction by finding common factors in the numerator and denominator to reduce them to their simplest form.
5. After simplifying each term, write the final expression as the sum and difference of the simplified terms.

Key Concepts

Distributive Property

The distributive property states that multiplying a sum or difference by a number is the same as multiplying each term inside the parentheses by that number separately, then adding or subtracting the results. For example, a(b + c) = ab + ac. This property is essential for rewriting and simplifying expressions involving sums or differences.

Fraction Multiplication and Simplification

When multiplying fractions, multiply the numerators together and the denominators together. Simplify the resulting fraction by dividing numerator and denominator by their greatest common divisor. This process helps in reducing complex fractional expressions to simpler forms.

Combining Like Terms

After distributing, terms with the same variable and exponent can be combined by adding or subtracting their coefficients. This simplification step reduces the expression to its simplest form, making it easier to interpret or use in further calculations.