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Chapter 2: Problem 13

Multiply. $$ 13(2 x+5) $$

Short Answer

Expert verified
The expression simplifies to \(26x + 65\).

Step by step solution

01

Distribute 13 to Each Term

To solve the problem, we first apply the distributive property: multiply 13 by each term inside the parentheses. The expression inside the parentheses is \(2x + 5\). We start by multiplying 13 by \(2x\), which gives us \(13 imes 2x = 26x\).
02

Multiply 13 by the Constant

Next, we multiply 13 by the constant term inside the parentheses, which is 5. This results in \(13 imes 5 = 65\).
03

Write the Simplified Expression

Finally, we combine the results from Step 1 and Step 2 to write the simplified expression. This gives us \(26x + 65\).

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Key Concepts

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

Multiplication in Algebra
In algebra, multiplication involves multiplying coefficients and variables. This is a key operation when working with algebraic expressions, such as those involving terms inside parentheses. Multiplying in algebra is similar to multiplication with numbers, but it requires careful attention to variables and proper use of algebraic rules.

When applying multiplication in algebra, you must:
  • Multiply the numerical coefficients together.
  • Multiply the variables, which often involves adding their exponents if they are the same.
In the given exercise, multiplication is applied through the distributive property, where 13 multiplies each term within the parentheses. This ensures every part of the expression is accounted for in the final product.
Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form, making it easier to understand and work with. You simplify by combining like terms, which are terms that have the same variable part.

When simplifying an expression:
  • Apply operations like addition, subtraction, multiplication, and division according to the rules of arithmetic.
  • Combine like terms, which are parts of the expression with the same variable and exponent.
After using the distributive property in the exercise, we arrived at an expression: \(26x + 65\). This expression is already in its simplest form because there are no like terms to combine. This step showcases the importance of identifying when an expression cannot be simplified further.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. It's a foundational concept in algebra that allows us to express generalized mathematical relationships.

Key components of algebraic expressions:
  • Constants: Numbers on their own (e.g., 5 in \(2x + 5\)).
  • Variables: Symbols that represent numbers (e.g., \(x\) in \(2x + 5\)).
  • Coefficients: Numbers in front of variables (e.g., 2 in \(2x\)).
Understanding each part of an algebraic expression helps you manipulate and solve expressions correctly. In the exercise, the expression \(2x + 5\) is transformed through multiplication, highlighting how variables and constants are treated separately to simplify or solve the problem.

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