Problem 72
Question
Simplify the expression. $$(x+8)(-2)-36$$
Step-by-Step Solution
Verified Answer
The simplified form of \((x+8)(-2)-36\) is \(-2x - 52.\)
1Step 1: Multiplication inside the brackets
First, let's do the multiplication inside the brackets. The brackets \((x+8)\) is to be multiplied with -2. So, \(-2*(x+8)\) equals \(-2x -16\). The expression now gets converted to \(-2x -16 - 36\).
2Step 2: Addition
Now, let's carry out the addition operation. -16 -36 equals to -52. So, the whole expression now simplifies to: \(-2x - 52\).
3Step 3: Final simplification
There are no further operations that we can carry out. Thus, the final simplified form of the given expression is: \(-2x-52\).
Key Concepts
Distributive PropertyCombining Like TermsBasic Algebra Operations
Distributive Property
The distributive property is a cornerstone of algebra which allows us to multiply a single term by each term within a set of parentheses. To put it simply, if you have an expression like \(a(b + c)\), the distributive property tells us that you can 'distribute' the multiplication of \(a\) across each term inside the parentheses, giving \(ab + ac\).
When we apply this to the given exercise \( (x+8)(-2) \) we distribute the \( -2 \) across both \( x \) and \( 8 \) to get \( -2x - 16 \) before subtracting \( 36 \) from both terms. This property is extremely useful when simplifying algebraic expressions, as it lays the groundwork for combining like terms and performing other basic algebra operations.
When we apply this to the given exercise \( (x+8)(-2) \) we distribute the \( -2 \) across both \( x \) and \( 8 \) to get \( -2x - 16 \) before subtracting \( 36 \) from both terms. This property is extremely useful when simplifying algebraic expressions, as it lays the groundwork for combining like terms and performing other basic algebra operations.
Combining Like Terms
Combining like terms is about merging terms that have the same variable raised to the same power. It's like organizing your groceries by putting all fruits together and all vegetables together; items are paired with their like kind.
Take, for example, our original expression after using the distributive property: \( -2x - 16 - 36 \). Here, \( -16 \) and \( -36 \) are like terms because they are both constants without any variables. To combine them, we simply add (or in this case, add the negative of) their coefficients to consolidate them into one term, which gives us \( -52 \). The resulting expression \( -2x - 52 \) is much more streamlined and easier to understand at a glance compared to the original.
Take, for example, our original expression after using the distributive property: \( -2x - 16 - 36 \). Here, \( -16 \) and \( -36 \) are like terms because they are both constants without any variables. To combine them, we simply add (or in this case, add the negative of) their coefficients to consolidate them into one term, which gives us \( -52 \). The resulting expression \( -2x - 52 \) is much more streamlined and easier to understand at a glance compared to the original.
Basic Algebra Operations
Basic algebra operations include addition, subtraction, multiplication, and division - the building blocks of algebra. In our expression, after applying the distributive property, we use addition to combine like terms. The process of simplification often involves a sequence of these operations to progressively streamline the expression.
Starting with \( -2x - 16 \), we subtract \( 36 \) to perform an addition operation (keeping in mind that subtracting a positive is the same as adding a negative). Understanding when and how to apply these operations is critical in simplifying expressions to their most reduced form, as we did to reach the final answer \( -2x - 52 \). Remember that subtraction and addition are often the last steps in simplification, following the distributive property and combination of like terms.
Starting with \( -2x - 16 \), we subtract \( 36 \) to perform an addition operation (keeping in mind that subtracting a positive is the same as adding a negative). Understanding when and how to apply these operations is critical in simplifying expressions to their most reduced form, as we did to reach the final answer \( -2x - 52 \). Remember that subtraction and addition are often the last steps in simplification, following the distributive property and combination of like terms.