Problem 40 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -9(a+6) $$

Step-by-Step Solution

Answer

-9a - 54

Step 1: Identify the expression to which Distributive Property is to be applied

In this expression, -9(a+6), -9 should be distributed over the sum of a and 6.

Step 2: Apply Distributive Property

By applying the distributive property, the expression -9(a+6) is equivalent to: -9*a + -9*6.

Step 3: Simplify the expression

Simplify the expression to obtain the final answer: -9a - 54

Key Concepts

Algebraic Expressions

Algebraic expressions are like short sentences in math that include numbers, variables, and operation symbols. For example, in the expression \(-9(a+6)\),

• \(-9\) is a number, known as a coefficient when combined with a variable.
• \(a\) is a variable, which stands for an unknown value you need to solve for.
• The operation inside the parentheses is addition.

In algebraic expressions, numbers and variables can be added, subtracted, multiplied, or divided according to algebra's rules. This particular expression involves multiplication of a number with a sum inside the parentheses. To handle this, we use the distributive property to rearrange and simplify the expression.

Simplification

Simplification in algebra involves reducing an expression to its simplest form. This means making it as compact as possible by performing operations like addition, subtraction, or multiplication.Applying the distributive property is a key step in simplification. In our example, we start with this expression \(-9(a+6)\).

• First, distribute the \(-9\) across each term inside the parentheses: \(-9 * a\) and \(-9 * 6\).
• After distributing, you get \(-9a - 54\).
• There are no like terms to combine, so your result \(-9a - 54\) is already simplified.

Simplification helps make expressions easier to work with, especially when solving equations or comparing results. Always look for opportunities to distribute numbers and combine like terms for clarity.

Negative Numbers

Negative numbers can sometimes seem tricky but are simply numbers less than zero. They play a critical role when simplifying expressions and applying the distributive property. In the expression \(-9(a+6)\):

• \(-9\) is a negative number, meaning it's less than zero.
• When you multiply a negative number by another number, it changes the sign of the product. For instance, \(-9 * a\) becomes \(-9a\) and remains negative since 'a' doesn't define positivity/negativity here directly.
• Similarly, \(-9 * 6\) results in \(-54\), another negative number.

Understanding how negative numbers behave is essential, particularly when dealing with algebraic expressions, as operations involving them follow specific rules. Using these rules correctly can help prevent errors and lead to accurate results. Keep practicing, and negative numbers will soon become second nature!