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Chapter 2: Problem 36

Multiply out and simplify as completely as possible. $$8(r-s+t)$$

Short Answer

Expert verified
8r - 8s + 8t

Step by step solution

01

Distribute the Multiplication

Multiply each term inside the parentheses by the coefficient outside the parentheses. In this case, distribute the 8 across each term inside the parentheses: \(8(r-s+t) = 8 \times r - 8 \times s + 8 \times t\).
02

Simplify Each Term

Perform the multiplications for each term: \(8 \times r = 8r\), \(8 \times (-s) = -8s\), and \(8 \times t = 8t\).
03

Combine the Terms

Combine the simplified terms into one expression: \(8r - 8s + 8t\).

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. They represent mathematical relationships and can be simplified or manipulated according to algebraic rules. In the exercise, the expression inside the parentheses, \(r - s + t\), is an example. Understanding algebraic expressions involves recognizing:
  • Variables like \(r\), \(s\), and \(t\).
  • Operation signs like addition (+) and subtraction (-).
  • Coefficients, which are numbers that multiply variables, like the 8 outside the parentheses.
By becoming comfortable with these elements, you can begin to work with more complex algebraic expressions.
Simplification
Simplification in algebra involves rewriting expressions in a more concise form. This often means combining like terms or eliminating parentheses through distribution. In our exercise, simplification begins with distributing the 8 to each term inside the parentheses:
\(8(r-s+t) \) becomes \(8r - 8s + 8t\).
Each term is simplified separately:
  • \(8 \times r = 8r\)
  • \(8 \times (-s) = -8s\)
  • \(8 \times t = 8t\)

After these multiplications, the expression is combined to give \(8r - 8s + 8t\). Simplification helps make expressions easier to understand and solve.
Multiplication in Algebra
Multiplication in algebra extends beyond simple numbers to include variables and entire expressions. In the given problem, we used the distributive property to multiply a single number across an expression in parentheses. The steps involved are:
  • Identify the number outside the parentheses (8, in this case).
  • Multiply this number by each term inside the parentheses: \(8 \times r, 8 \times (-s), 8 \times t\).
  • Perform the multiplication to simplify each term individually: \(8r, -8s, 8t\).

Using these steps, the expression \(8(r-s+t)\) simplifies to \(8r - 8s + 8t\). By understanding and applying these principles of multiplication in algebra, solving similar problems becomes straightforward.

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