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Chapter 7: Problem 91

In the following exercises, simplify using the distributive property. $$3(a+9)$$

Short Answer

Expert verified
3(a + 9) simplifies to 3a + 27.

Step by step solution

01

- Understand the Distributive Property

In the distributive property, you multiply the term outside the parentheses by each term inside the parentheses separately. The property is expressed as: \[ a(b + c) = ab + ac \].
02

- Apply the Distributive Property

To simplify \[ 3(a + 9) \], multiply the term outside the parentheses (3) by each term inside the parentheses (a and 9) separately. This gives us: \[ 3 \times a + 3 \times 9 \].
03

- Simplify Each Term

Next, perform the multiplications: \[ 3 \times a = 3a \] and \[ 3 \times 9 = 27 \].
04

- Combine the Results

Add the results together to get the simplified expression: \[ 3a + 27 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions is a crucial skill in algebra. It involves making an expression easier to work with by combining like terms and using mathematical properties such as the distributive property. When simplifying, always look for opportunities to reduce the expression to its most basic form. This helps in solving equations more efficiently and accurately. For example, in the expression \(3(a + 9)\), we used the distributive property to break down the terms inside the parentheses. Remember to always perform operations step-by-step to avoid mistakes.
Algebraic Multiplication
Algebraic multiplication involves multiplying variables and constants. When dealing with parentheses, like in our example \(3(a + 9)\), use the distributive property: multiply the outside term (3) by each inside term (a and 9). This process is written as:

\[ 3 \times a + 3 \times 9 \].

Ensure each term is multiplied carefully:

  • First, multiply 3 by a to get \(3a\).
  • Next, multiply 3 by 9 to get 27.

Combining these results gives us \(3a + 27\). Breaking down the multiplication helps in avoiding errors and makes the process more manageable.
Basic Algebra
Understanding basic algebra is the foundation for more complex math concepts. It involves working with numbers and variables to solve equations and simplify expressions. The key principles include:

  • Understanding variables as symbols that represent numbers.
  • Knowing the order of operations (PEMDAS/BODMAS) - Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
  • Using properties like the distributive property: \(a(b + c) = ab + ac\).

In our example, we applied the distributive property to simplify \(3(a + 9)\) to \(3a + 27\). Mastery of these basic concepts is essential, as they are frequently used in solving algebraic problems.

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