Problem 41
Question
For the following exercises, simplify the expression. \(8 b-4 b(3)+1\)
Step-by-Step Solution
Verified Answer
The simplified expression is \(-4b + 1\).
1Step 1: Distribute the Term Inside the Parentheses
Identify the term with parentheses, which is \(-4b(3)\). Distribute the multiplication through the parentheses: \(-4b \times 3 = -12b\).
2Step 2: Replace Distributed Expression in Original
Substitute \(-12b\) back into the expression, replacing \(-4b(3)\). The expression becomes: \(8b - 12b + 1\).
3Step 3: Combine Like Terms
Combine the terms with \(b\): \(8b - 12b = -4b\). Thus, the expression simplifies to: \(-4b + 1\).
Key Concepts
Distributive PropertyCombining Like TermsAlgebraic Expressions
Distributive Property
The distributive property is a fundamental concept in algebra that serves as a tool for simplifying expressions. It allows you to multiply a single term with the terms within parentheses. This property can be thought of as distributing the multiplication across each term in the brackets separately.
In the exercise, the term \( -4b(3) \) applies the distributive property by multiplying \( -4b \) with \( 3 \), resulting in \( -12b \). Distributing like this helps to eliminate parentheses and prepare the expression for further simplification. This technique is crucial when dealing with algebraic expressions, as it simplifies the problem by breaking it down into more manageable parts.
Remember, the distributive property applies as follows:
In the exercise, the term \( -4b(3) \) applies the distributive property by multiplying \( -4b \) with \( 3 \), resulting in \( -12b \). Distributing like this helps to eliminate parentheses and prepare the expression for further simplification. This technique is crucial when dealing with algebraic expressions, as it simplifies the problem by breaking it down into more manageable parts.
Remember, the distributive property applies as follows:
- For any numbers or variables \( a \), \( b \,\) and \( c \, \) the formula is: \( a(b + c) = ab + ac \).
- This means you multiply \( a \) with each term inside the parentheses separately.
Combining Like Terms
After applying the distributive property, you often deal with terms that can be combined. The process of "combining like terms" is about merging terms that have the same variables and corresponding powers, allowing for further streamlining of an expression.
In the expression from our example, \( 8b - 12b + 1 \, \) you can see two terms with \( b \) in them: \( 8b \) and \( -12b \, \) which makes them like terms. Combining these, \( 8b - 12b \), gives \( -4b \). This step reduces the expression to a simpler form, making it much easier to handle!
In the expression from our example, \( 8b - 12b + 1 \, \) you can see two terms with \( b \) in them: \( 8b \) and \( -12b \, \) which makes them like terms. Combining these, \( 8b - 12b \), gives \( -4b \). This step reduces the expression to a simpler form, making it much easier to handle!
- Like terms are terms that include the same variables raised to identical powers.
- For example, \( 2x \) and \( 5x \) can be combined to become \( 7x \, \) whereas \( 2x \) and \( 2y \) cannot be combined.
Algebraic Expressions
Understanding algebraic expressions is a cornerstone of learning algebra. An algebraic expression consists of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). These expressions do not have an equality sign, which differentiates them from algebraic equations.
In our exercise, the expression is \( 8b - 4b(3) + 1 \, \) where \( b \) is the variable. Simplifying an algebraic expression involves using properties like the distributive property and combining like terms to transform the expression into its simplest form.
Some key points to remember:
In our exercise, the expression is \( 8b - 4b(3) + 1 \, \) where \( b \) is the variable. Simplifying an algebraic expression involves using properties like the distributive property and combining like terms to transform the expression into its simplest form.
Some key points to remember:
- Variables represent unknown values and can change depending on the conditions of the problem.
- Operational signs like \( +, -, imes \), and \( \div \) guide how to manipulate these expressions.
- The goal is to make the expression as concise as possible, without changing its fundamental value.
Other exercises in this chapter
Problem 41
For the following exercises, simplify each expression. \(3 \sqrt{a b^{2}}-b \sqrt{a}\)
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For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(b^{-3} c\right)^{3}\)
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For the following exercises, simplify the rational expression. \(\frac{\frac{6}{y}-\frac{4}{x}}{y}\)
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For the following exercises, factor the polynomials. \(729 q^{3}+1331\)
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