Problem 10
Question
Fill in the ( ) to make each statement true. Use the associative properties. \([(7 m-2)(m+3)](m+4)=(7 m-2)[(\quad)(\quad)]\)
Step-by-Step Solution
Verified Answer
Question: Fill in the blank spaces with the correct parenthesis grouping based on the associative property: \([(7 m-2)(m+3)](m+4)=(7 m-2)[(\quad)(\quad)]\)
Answer: \([(7 m-2)(m+3)](m+4)=(7 m-2)[(m+3)(m+4)]\)
1Step 1: Set Up the given equation
We are given the following equation:
\([(7 m-2)(m+3)](m+4)=(7 m-2)[(\quad)(\quad)]\)
We need to fill in the blank spaces with the correct parenthesis grouping such that the associative property holds true.
2Step 2: Apply the associative property of multiplication
In order to apply the associative property, the numbers can be regrouped while keeping the same number of parentheses.
Since we need to equate both sides of the given equation, we can rewrite the left side of the equation by regrouping the expressions within the brackets as follows:
\([(7 m-2)(m+3)](m+4) = (7 m-2)((m+3)(m+4))\)
3Step 3: Write the final equation with the correct grouping
After applying the associative property, we can now fill in the blank spaces in the given equation with the correct grouping terms. We have:
\([(7 m-2)(m+3)](m+4)=(7 m-2)[(m+3)(m+4)]\)
Now, the equation is complete and the associative property holds true with the correct grouping of parenthesis.
Key Concepts
Elementary AlgebraEquation SolvingMultiplication Grouping
Elementary Algebra
Elementary algebra is the foundation of all mathematical expressions that involve variables and equations. It allows us to deal with mathematical symbols and represent problem-solving processes.
In this level of algebra, we learn to form equations and know how to solve them using set rules and properties, such as the associative property. This property is particularly useful when dealing with expressions that include operations like addition and multiplication.
Elementary algebra introduces concepts like variables, which are symbols (often letters) representing numbers. For example, in the expression \(7m - 2\), \(m\) is a variable that can take on different values. By mastering elementary algebra, students learn to manipulate these variables and understand how to form and solve equations.
In this level of algebra, we learn to form equations and know how to solve them using set rules and properties, such as the associative property. This property is particularly useful when dealing with expressions that include operations like addition and multiplication.
Elementary algebra introduces concepts like variables, which are symbols (often letters) representing numbers. For example, in the expression \(7m - 2\), \(m\) is a variable that can take on different values. By mastering elementary algebra, students learn to manipulate these variables and understand how to form and solve equations.
Equation Solving
Equation solving is an essential skill in algebra that involves finding values for variables that make an equation true. It requires an understanding of various algebraic properties and rules, such as the associative, commutative, and distributive properties.
To solve equations, you often need to manipulate them using these properties so that you can isolate the variable and find its value. This can involve:
To solve equations, you often need to manipulate them using these properties so that you can isolate the variable and find its value. This can involve:
- Combining like terms
- Balancing equations by performing the same operation on both sides
- Applying algebraic identities and properties
Multiplication Grouping
Multiplication grouping utilizes the associative property of multiplication, which states that how numbers are grouped in a multiplication operation doesn't affect the final result. In algebraic terms, the associative property is expressed as \((a \times b) \times c = a \times (b \times c)\).
This is especially useful in handling expressions with multiple factors, allowing us to simplify or solve them more easily. It involves grouping diagrams with parentheses to indicate which operations to perform first. Think of it as akin to stacking blocks, where the order in which you stack them doesn't change the height of the stack.
In the equation from the exercise, the associative property allows us to regroup the multiplication as \((7m-2)(m+3)(m+4) = (7m-2)((m+3)(m+4))\). This regrouping helps in simplifying the problem, as now the expressions inside the grouped parentheses can be dealt with easily.
This is especially useful in handling expressions with multiple factors, allowing us to simplify or solve them more easily. It involves grouping diagrams with parentheses to indicate which operations to perform first. Think of it as akin to stacking blocks, where the order in which you stack them doesn't change the height of the stack.
In the equation from the exercise, the associative property allows us to regroup the multiplication as \((7m-2)(m+3)(m+4) = (7m-2)((m+3)(m+4))\). This regrouping helps in simplifying the problem, as now the expressions inside the grouped parentheses can be dealt with easily.
Other exercises in this chapter
Problem 10
Make use of either or both the power rule for products and the power rule for powers to simplify each expression. $$ \left[10 t^{4} y^{7} j^{3} d^{2} v^{6} n^{4
View solution Problem 10
Use the order of operations to simplify the following. $$ 1^{4}+\left(2^{2}+4\right)^{2} \div 2^{3} $$
View solution Problem 10
Is there a largest positive number? Is there a smallest negative number?
View solution Problem 10
For the following problems, use the order of operations to find each value. $$2+3(6)$$
View solution