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Chapter 2: Problem 44

Multiply out and simplify as completely as possible. $$-8(5 r+6 t-4 u)$$

Short Answer

Expert verified
-40r - 48t + 32u

Step by step solution

01

Distribute the -8

Apply the distributive property by multiplying each term inside the parentheses by -8.
02

Multiply each term

Multiply \(5r\) by \(-8\)\[ -8(5r) = -40r \] Multiply \(6t\) by \(-8\)\[ -8(6t) = -48t \] Multiply \(-4u\) by \(-8\)\[ -8(-4u) = 32u \]
03

Combine the terms

Combine the multiplied terms to obtain the simplified expression:\[ -40r - 48t + 32u \]

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Key Concepts

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

Multiplication of Terms
In algebra, the multiplication of terms involves multiplying constants and variables from one set to another. This is essential when using the distributive property. As shown in the exercise, we multiplied \(-8\) by each term inside the parentheses: \(5r\), \(6t\), and \(-4u\). This resulted in:
  • Multiplying \(5r\) by \(-8\) to get \(-40r\)
  • Multiplying \(6t\) by \(-8\) to get \(-48t\)
  • Multiplying \(-4u\) by \(-8\) to get \(32u\)
Each multiplication step follows the principle of multiplying the coefficients (the numerical part) and carrying along the variable part.
Simplifying Expressions
Simplifying expressions means combining like terms and making the expression as concise as possible.
In the given exercise, after multiplying out the terms, you should end up with \(-40r - 48t + 32u\).
This is a simplified expression.

Steps to simplify:
  • First, distribute any constants through every term inside the parentheses.
  • Next, perform the multiplication.
  • If possible, combine like terms (terms with the same variable and exponent).
  • Ensure there are no further combinations or unnecessary terms.
Here, \(-40r\), \(-48t\), and \(32u\) are all different terms, so they cannot be further combined.
Algebraic Operations
Algebraic operations involve basic mathematical operations like addition, subtraction, multiplication, and division using algebraic expressions.
In our example, we focused on multiplication and addition/subtraction through the application of the distributive property.
Steps involved:
  • Multiplication: Each term inside the parentheses was multiplied by the \(-8\), the coefficient outside.
  • Combining: Once all terms were multiplied, they were combined together into a single expression, ensuring each term was simplified and no like terms were further combined.
  • The Distributive Property: This key algebraic principle states that \(a(b + c + d) = ab + ac + ad\). It allows us to multiply a single term (like \(-8\)) to each term within parentheses efficiently.
Mastering these operations is crucial for solving complex algebraic problems.

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