Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$-9(3 x)$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression \( -9(3x) \) is \( -27x \)

Step 1: Identify the Operations

In this case, the operation is multiplication. The term -9 is multiplied by the term inside the parentheses, which is 3x.

Step 2: Apply Distributive Property

Proceed to multiply -9 with 3x. The distributive property states that for all \( a \), \( b \), and \( c \), the equation \( a(b + c) = ab + ac \) holds. Although there is no explicit addition in this case, this law still applies.

Step 3: Perform the Multiplication

Perform the multiplication operation to simplify the expression. \( -9 \) times \( 3x \) gives \( -27x \).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows one to distribute multiplication over addition or subtraction within parentheses. For example, when given the expression \( a(b + c) \), the distributive property lets us 'break' this compound term into two simpler terms, \( ab + ac \). This is essential when you want to simplify an expression or to multiply a single term by a polynomial.

While the original exercise \( -9(3x) \) does not present an addition or subtraction inside the parentheses, this property applies because it’s an implicit multiplicative distribution of a single number over a term. To simplify \( -9(3x) \) using the distributive property, you'd directly multiply \( -9 \) by each term inside the parentheses; in this instance, only \( 3x \) is present. It is vital always to watch for negative signs to avoid mistakes, leading to \( -27x \) after distribution.

Multiplication in Algebra

Multiplication in algebra may seem daunting, but it follows a straightforward principle: when multiplying constants and variables, you directly multiply the coefficients (numerical parts) and apply the appropriate exponent rules for the variables, if needed. So for constants like \( -9 \) and \( 3 \), you would simply multiply these numbers together.

Variables like \( x \) carry through the multiplication. Hence, when you come across \( -9 \) times \( 3x \), you multipl \( -9 \) and \( 3 \) to obtain \( -27 \) and the \( x \) is carried over as it is not being multiplied by another variable. It's important to remember that the sign of the product depends on the signs of the terms being multiplied. Since one is negative and the other is positive here, the result \( -27x \) is negative.

Intermediate Algebra

Intermediate algebra involves extending the principles of basic algebra to more complex problems often including polynomials, rational expressions, and systems of equations. The objective is to develop proficiency in manipulating algebraic expressions and solving equations adeptly.

Understanding how to simplify algebraic expressions, such as \( -9(3x) \) effectively, is critical in intermediate algebra. This level involves recognizing and applying different properties of algebra, such as the distributive property, to find solutions. Simplification aids in easier manipulation of terms and often precedes more advanced steps like factorization or solving equations. For the effective learning of intermediate algebra, it's recommended to practice the foundational skills, like multiplication and distribution, as seen in the exercise. These skills form the building blocks for solving more complicated algebraic problems.