Problem 146
Question
For the following problems, write each expression so that only positive exponents appear. Find the difference. \(8-(-12)\).
Step-by-Step Solution
Verified Answer
Question: Find the difference between the given numbers: 8 - (-12).
Answer: The difference between the numbers is 20.
1Step 1: Understand the given expression
Given expression: \(8-(-12)\). We need to find the difference between the numbers.
2Step 2: Solve the expression
To find the difference, we must change the subtraction of a negative number into the addition of its corresponding positive number:
\(8-(-12)\) becomes \(8+12\).
Now, add the numbers:
\(8+12 = 20\).
3Step 3: Write the final answer
The difference between the numbers is \(20\).
Key Concepts
Understanding Algebraic ExpressionsSubtracting Negative NumbersSimplifying Expressions
Understanding Algebraic Expressions
An algebraic expression represents a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (such as addition, subtraction, multiplication, and division). For example, in the expression \(8-(-12)\), '8' and '-12' are the numerical components, while the minus sign ' - ' shows a subtraction operation.
The beauty of algebraic expressions lies in their ability to represent real-world situations in an abstract way, allowing us to work with unknown or variable quantities systematically. Simplifying these expressions is an essential skill in algebra and helps students transform complex or confusing terms into easier, more digestible chunks of information.
To further grasp the concept, let's look at the simple algebraic expression of \(a+b\). If \(a=3\) and \(b=5\), substituting these values into the expression yields \(3+5\), or '8', a much simpler figure to work with.
The beauty of algebraic expressions lies in their ability to represent real-world situations in an abstract way, allowing us to work with unknown or variable quantities systematically. Simplifying these expressions is an essential skill in algebra and helps students transform complex or confusing terms into easier, more digestible chunks of information.
To further grasp the concept, let's look at the simple algebraic expression of \(a+b\). If \(a=3\) and \(b=5\), substituting these values into the expression yields \(3+5\), or '8', a much simpler figure to work with.
Subtracting Negative Numbers
When it comes to subtracting negative numbers, it often creates confusion. The reason is that subtracting a negative can feel counterintuitive until we realize the rule: subtracting a negative number is equivalent to adding its positive counterpart.
For instance, consider our original problem \(8-(-12)\). To make sense of this, remember that two negatives make a positive; hence, \(8-(-12)\) becomes \(8+12\), which clearly shows we are adding 12 to 8, not taking it away.
Understanding this concept is crucial not only for pure math but also for real-world scenarios like financial transactions, where 'subtracting a debt' essentially means 'adding credit' or increasing one's net worth.
For instance, consider our original problem \(8-(-12)\). To make sense of this, remember that two negatives make a positive; hence, \(8-(-12)\) becomes \(8+12\), which clearly shows we are adding 12 to 8, not taking it away.
Understanding this concept is crucial not only for pure math but also for real-world scenarios like financial transactions, where 'subtracting a debt' essentially means 'adding credit' or increasing one's net worth.
Simplifying Expressions
Simplifying expressions is a foundational skill in algebra that involves reducing an expression to its simplest form. This process makes calculations easier and helps clarify solutions. In our example, simplifying the expression \(8-(-12)\) involves eliminating the negative sign before the 12 to reduce the expression to \(8+12\), a much straightforward addition problem.
Simplification can involve combining like terms, factoring, expanding expressions, or, as shown, changing the signs of numbers. The key is to transform the expression so that it clearly conveys the same information in a simpler manner. Always performing these simplifications systematically ensures that you won't miss any opportunity to make an expression as easy to understand as possible.
In summary, simplifying reduces potential errors, promotes understanding, and leads to clearer, more immediate calculations.
Simplification can involve combining like terms, factoring, expanding expressions, or, as shown, changing the signs of numbers. The key is to transform the expression so that it clearly conveys the same information in a simpler manner. Always performing these simplifications systematically ensures that you won't miss any opportunity to make an expression as easy to understand as possible.
In summary, simplifying reduces potential errors, promotes understanding, and leads to clearer, more immediate calculations.
Other exercises in this chapter
Problem 144
For the following problems, write each expression so that only positive exponents appear. Simplify \(\left(4 x^{5} y^{3} z^{0}\right)^{3}\)
View solution Problem 145
For the following problems, write each expression so that only positive exponents appear. Find the sum. \(-15+3\).
View solution Problem 147
For the following problems, write each expression so that only positive exponents appear. Simplify \((-3)(-8)+4(-5)\)
View solution Problem 148
For the following problems, write each expression so that only positive exponents appear. Find the value of \(m\) if \(m=\frac{-3 k-5 t}{k t+6}\) when \(k=4\) a
View solution