Problem 78
Question
Simplify each numerical expression. $$-2(-7+13)+6(-3-2)$$
Step-by-Step Solution
Verified Answer
The simplified expression is -42.
1Step 1: Simplify inside the parentheses
Start by performing the operations inside the parentheses. For (-7 + 13), calculate the addition: \(-7 + 13 = 6\). For (-3 - 2), calculate the subtraction: \(-3 - 2 = -5\). Now the expression becomes: \(-2(6) + 6(-5)\).
2Step 2: Apply the distributive property
Now that we have simple expressions inside the parentheses, use the distributive property to eliminate them. For \(-2(6)\), multiply \(-2\) by \(6\) to get \(-12\). Similarly, for \(6(-5)\), multiply \(6\) by \(-5\) to get \(-30\). Now the expression becomes: \(-12 + (-30)\).
3Step 3: Simplify the expression
Finally, combine the terms to simplify the expression. Since both terms are negative, add their absolute values and keep the negative sign: \(-12 + (-30) = -42\).
Key Concepts
Distributive propertyNumerical expressionsOrder of operations
Distributive property
The distributive property is an essential tool in algebra that helps us to simplify expressions involving multiplication and addition or subtraction. In this property, you distribute a factor across terms inside a set of parentheses. This means multiplying each term inside the parentheses individually by the factor outside.
For example, if we have an expression like \(a(b + c)\), the distributive property allows us to rewrite it as \(ab + ac\).
In the original exercise, we use the distributive property twice. First, with \(-2(6)\) where we multiply \(-2\) by \(6\), resulting in \(-12\). And then with \(6(-5)\), multiplying \(6\) by \(-5\) to get \(-30\).
Applying this property helps break down complex expressions into simpler components, making them easier to work with and solve.
For example, if we have an expression like \(a(b + c)\), the distributive property allows us to rewrite it as \(ab + ac\).
In the original exercise, we use the distributive property twice. First, with \(-2(6)\) where we multiply \(-2\) by \(6\), resulting in \(-12\). And then with \(6(-5)\), multiplying \(6\) by \(-5\) to get \(-30\).
Applying this property helps break down complex expressions into simpler components, making them easier to work with and solve.
Numerical expressions
Numerical expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division, but they don't include any variables. They often look like statements that need simplification, as we can see in this exercise.
Here, we begin with \(-2(-7+13)+6(-3-2)\), a numerical expression where each operation needs to be performed in the correct sequence using the rules of mathematics. Once operations within parentheses are simplified, what remains can further be simplified using other properties or rules.
By simplifying numerical expressions accurately, it ensures that calculations and results maintain consistency and correctness, which is crucial in solving math problems.
Here, we begin with \(-2(-7+13)+6(-3-2)\), a numerical expression where each operation needs to be performed in the correct sequence using the rules of mathematics. Once operations within parentheses are simplified, what remains can further be simplified using other properties or rules.
By simplifying numerical expressions accurately, it ensures that calculations and results maintain consistency and correctness, which is crucial in solving math problems.
Order of operations
The order of operations is a set of rules that dictate the correct sequence to evaluate a mathematical expression. This ensures everyone reaches the same result by performing operations in the same order. The standard order is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In the exercise, following the order of operations, we first simplify the calculations inside the parentheses: \(-7+13\) and \(-3-2\). These simplify to \(6\) and \(-5\), respectively.
After that, we apply multiplication which follows, using the distributive property as well. Finally, after all multiplication and distribution, we perform the last operation, which in this case, is addition of \(-12\) and \(-30\), resulting in \(-42\). By strictly following the order of operations, complex numerical expressions can be simplified correctly and efficiently.
In the exercise, following the order of operations, we first simplify the calculations inside the parentheses: \(-7+13\) and \(-3-2\). These simplify to \(6\) and \(-5\), respectively.
After that, we apply multiplication which follows, using the distributive property as well. Finally, after all multiplication and distribution, we perform the last operation, which in this case, is addition of \(-12\) and \(-30\), resulting in \(-42\). By strictly following the order of operations, complex numerical expressions can be simplified correctly and efficiently.
Other exercises in this chapter
Problem 77
Your friend keeps getting an answer of 30 when simplifying \(7+8(2)\). What mistake is he making and how would you help him?
View solution Problem 78
Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. Twelve times the sum of a number and 7
View solution Problem 78
Do you think \(3 \sqrt{2}\) is a rational or an irrational number? Defend your answer.
View solution Problem 79
Answer the question with an algebraic expression. Brian is \(n\) years old. How old will he be in 20 years?
View solution