Problem 2
Question
\(5(6 x)\) and \(30 x\) are ______ expressions because for each value of \(x,\) they represent the same number.
Step-by-Step Solution
Verified Answer
Equivalent expressions.
1Step 1: Understanding the Problem
The problem is asking us to identify the type of mathematical expressions that are equivalent for every value of \(x\). We are given two expressions: \(5(6x)\) and \(30x\).
2Step 2: Simplifying the First Expression
The first expression is \(5(6x)\). We can simplify this by using the distributive property: \(5 \times 6 \times x\). Simplifying further, we get \(30x\).
3Step 3: Comparing the Expressions
The expression \(5(6x)\) simplifies to \(30x\). The second expression is already \(30x\). Therefore, both expressions simplify to the same form.
4Step 4: Identifying the Type of Expression
Since \(5(6x)\) and \(30x\) simplify to the same expression for every value of \(x\), they are called 'equivalent expressions'. Equivalent expressions represent the same quantity regardless of the value of the variable.
Key Concepts
Distributive PropertySimplifying ExpressionsMathematical Expressions
Distributive Property
The distributive property is a fascinating and fundamental principle in mathematics that assists in simplifying expressions and makes complex calculations more manageable. It states that multiplying a number by a sum gives the same result as multiplying that number by each addend separately and then adding the products. In mathematical terms, it can be expressed as \(a(b + c) = ab + ac\).
This property is helpful when we deal with expressions like \(5(6x)\) because it enables us to break down the multiplication into more straightforward parts. By applying the distributive property, we first multiply 5 by 6 to get 30, and then we multiply by \(x\), resulting in \(30x\). This transformation shows how the distributive property can turn a slightly more complicated expression into one that is easier to handle.
Understanding and using the distributive property is crucial in algebra and beyond, as it extends to various math and real-life applications. Its ability to simplify and reorganize expressions is a key tool in mathematical problem-solving.
This property is helpful when we deal with expressions like \(5(6x)\) because it enables us to break down the multiplication into more straightforward parts. By applying the distributive property, we first multiply 5 by 6 to get 30, and then we multiply by \(x\), resulting in \(30x\). This transformation shows how the distributive property can turn a slightly more complicated expression into one that is easier to handle.
Understanding and using the distributive property is crucial in algebra and beyond, as it extends to various math and real-life applications. Its ability to simplify and reorganize expressions is a key tool in mathematical problem-solving.
Simplifying Expressions
Simplifying expressions is a process where we reduce an expression to its simplest form without changing its value. The goal is to make the expression easier to read, interpret, and manipulate. In our example, we are given two expressions: \(5(6x)\) and \(30x\). The exercise demonstrates how simplifying \(5(6x)\) using the distributive property hints at their equivalence.
When we simplify \(5(6x)\), we effectively rewrite it as \(30x\), showing both expressions are identical in value. This simplified form allows for easier comparison and further operations within equations or real-world scenarios.
Key elements to remember when simplifying include:
When we simplify \(5(6x)\), we effectively rewrite it as \(30x\), showing both expressions are identical in value. This simplified form allows for easier comparison and further operations within equations or real-world scenarios.
Key elements to remember when simplifying include:
- Identifying like terms, which are terms containing the same variable raised to the same power.
- Using mathematical properties, such as the associative or distributive property, to combine terms efficiently.
Mathematical Expressions
Mathematical expressions are combinations of numbers, variables, and operations that represent a specific value. They form the backbone of algebra and are integral to conveying mathematical ideas and relationships. In our exercise, the expressions \(5(6x)\) and \(30x\) are examples of expressing quantities and relationships in a versatile and concise way.
Mathematical expressions can include:
Understanding expressions helps develop a strong foundation in mathematics as they can describe complex scenarios in an abstract and logical manner. By mastering the use and manipulation of expressions, one can solve equations, model real-world situations, and further mathematical understanding significantly.
Mathematical expressions can include:
- Variables, which serve as placeholders for unknown or varying values.
- Constants, which are fixed numbers.
- Operations, such as addition, subtraction, multiplication, and division.
Understanding expressions helps develop a strong foundation in mathematics as they can describe complex scenarios in an abstract and logical manner. By mastering the use and manipulation of expressions, one can solve equations, model real-world situations, and further mathematical understanding significantly.
Other exercises in this chapter
Problem 1
Fill in the blanks. The set of ___ numbers is \(\\{0,1,2,3,4,5, \ldots\\}.\)
View solution Problem 1
Fill in the blanks. A _____ is the result of an addition. A _____ of a subtraction. A _____ is the result of a multiplication. A ____ is the result of a divisio
View solution Problem 2
Fill in the blanks. A _____ is a product or quotient of numbers and/or variables. Examples are: \(8 x, \frac{t}{2},\) and \(-c d^{3}\)
View solution Problem 2
\(10^{2}\) can be read as ten_______ \(\quad,\) and \(10^{3}\) can be read as ten______.
View solution