Problem 57 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$z(x+9 w)$$

Step-by-Step Solution

Answer

Question: Use the distributive property to expand the expression $$z(x+9w)$$. Answer: $$z(x+9w) = zx + 9zw$$.

Step 1: Identify the variables and constants

In this expression, we have z as the variable to distribute and x and 9w as the variables inside the parentheses.

Step 2: Apply the distributive property

Using the distributive property, we multiply z by each term inside the parentheses and add the results together: $$z(x + 9w) = z \cdot x + z \cdot 9w$$

Step 3: Simplify the expression

Simplify the expression by performing the multiplication: $$zx + z \cdot 9w$$ $$zx + 9zw$$ The expanded expression using the distributive property is: $$z(x+9w) = zx + 9zw$$

Key Concepts

Algebraic Expression Expansion

Expanding algebraic expressions is a fundamental skill in algebra. It involves rewriting an expression in an equivalent form without parentheses, which often reveals its underlying structure. For example, take the simple algebraic expression \( z(x + 9w) \). Expanding this requires applying the distributive property, which leads to the expression being rewritten as \( zx + 9zw \).

The act of expansion can transform a compact expression into a form that is more suitable for further algebraic manipulations such as factorisation or simplification. Whenever you encounter parentheses in an algebraic expression and need to expand, look for opportunities to distribute any multiplication outside the parentheses across the terms within.

Applying the Distributive Property

The distributive property is a cornerstone in algebra that allows us to simplify expressions. It states that \( a(b + c) = ab + ac \), where \( a \) is multiplied by each term inside the parentheses. When we apply the distributive property to an expression like \( z(x + 9w) \), we distribute the \( z \) across the \( x \) and \( 9w \) to obtain \( zx + 9zw \).

By mastering the distributive property, algebraic expressions that initially seem complex can be expanded and simplified with ease. This process is like sharing something evenly among friends—\( z \) is shared across \( x \) and \( 9w \) to ensure each term gets its fair share.

Simplifying Expressions

Once the distributive property has been applied, the next step involves simplifying the expression to its most basic form. This means combining like terms, reducing coefficients, and in some cases, factoring. With our example, after distributing \( z \), we end up with \( zx + 9zw \), which is already in its simplest form.

Simplification makes expressions clearer and often easier to work with, especially when solving equations or evaluating expressions for different values of the variables. Remember, a simplified expression should have no parentheses, no like terms that can be combined, and all coefficients should be reduced to their lowest terms.

Multiplication of Algebraic Terms

Multiplying algebraic terms is another key aspect of dealing with expressions. To multiply terms, multiply the coefficients (the numerical parts) and add the exponents of like variables. In the expression \( zx + 9zw \), we're multiplying \( z \) by \( x \) and \( z \) by \( 9w \).

For example, if we had to multiply \( z \) and \( x^2 \) instead, the product would be \( zx^3 \) (provided \( z \) and \( x \) are not like terms). Understanding how to correctly multiply terms ensures you can expand algebraic expressions correctly and is crucial for more advanced algebraic operations.