Problem 12
Question
Find the opposite of each real number. $$ -[-(-7)] $$
Step-by-Step Solution
Verified Answer
Answer: The opposite of the given real number in the expression is \(7\).
1Step 1: Identify and Simplify the Innermost Parentheses
Start by looking at the innermost parentheses, which is \((-7)\). The negative sign in front of it means to find the opposite of \(-7\), which is \(7\). Now the expression becomes \(-[-(7)]\).
2Step 2: Simplify the Next Parentheses and Add Negative Sign
Now, look at the updated innermost parentheses, which is \((7)\). The negative sign in front of it means to find the opposite of \(7\), which is \(-7\). So the expression becomes \(-[-7]\).
3Step 3: Simplify the Last Remaining Parentheses
The last remaining parentheses is \((-7)\). The negative sign in front of it means to find the opposite of \(-7\), which is \(7\). So the final expression is \(7\).
So, the opposite of the given real number is \(7\).
Key Concepts
Understanding Real NumbersSimplifying Expressions with NegativesGrasping Negative Numbers
Understanding Real Numbers
Real numbers are the collection of all rational and irrational numbers, which include integers, fractions, and decimals that can represent a distance along a continuous line, often referred to as the number line. For example, numbers like 2, -3, 0.75, and \(\sqrt{2}\) are real numbers. This set does not include imaginary or complex numbers.
When we consider the opposite of a real number, we refer to the number that is the same distance from zero on the number line but in the opposite direction. For instance, the opposite of 3 is -3, and vice versa. The concept of opposites is integral when dealing with real numbers because it helps in simplifying expressions, especially when negatives are involved.
When we consider the opposite of a real number, we refer to the number that is the same distance from zero on the number line but in the opposite direction. For instance, the opposite of 3 is -3, and vice versa. The concept of opposites is integral when dealing with real numbers because it helps in simplifying expressions, especially when negatives are involved.
Simplifying Expressions with Negatives
Simplifying expressions is a fundamental skill in mathematics that involves reducing complex equations or expressions into a more straightforward and understandable form. This often includes combining like terms, removing parentheses, and simplifying fractions. When we encounter negative numbers in expressions, there are specific rules to follow.
Considering double negatives, we understand that two negatives make a positive. For instance, the expression \( -(-a) \) simplifies to \( a \), since the opposite of the opposite of a number returns us to our original number.
When simplifying expressions with multiple layers of negatives and parentheses, like in our exercise \( -[-(-7)] \), we must carefully evaluate each layer, working from the inside out to ensure accurate simplification.
Considering double negatives, we understand that two negatives make a positive. For instance, the expression \( -(-a) \) simplifies to \( a \), since the opposite of the opposite of a number returns us to our original number.
When simplifying expressions with multiple layers of negatives and parentheses, like in our exercise \( -[-(-7)] \), we must carefully evaluate each layer, working from the inside out to ensure accurate simplification.
Grasping Negative Numbers
Negative numbers are integers less than zero and they are typically represented with a minus sign (-) in front of them. They are essential when representing values such as debts, temperatures below freezing, or any quantity that falls below a defined zero point.
In mathematics, the concept of negative numbers is used to define the opposites of positive numbers. If we have a positive number 'a', its opposite is '-a', and the opposite of '-a' is 'a'. This is important in understanding the relationship between numbers on the number line and how they interact during addition, subtraction, multiplication, and division.
Moreover, when finding the opposite of the opposite of a number, we end up back at our starting number, reflecting the idea discussed in the previous sections. For example, the opposite of the opposite of -7 is -(-(-7)) which simplifies back to 7.
In mathematics, the concept of negative numbers is used to define the opposites of positive numbers. If we have a positive number 'a', its opposite is '-a', and the opposite of '-a' is 'a'. This is important in understanding the relationship between numbers on the number line and how they interact during addition, subtraction, multiplication, and division.
Moreover, when finding the opposite of the opposite of a number, we end up back at our starting number, reflecting the idea discussed in the previous sections. For example, the opposite of the opposite of -7 is -(-(-7)) which simplifies back to 7.
Other exercises in this chapter
Problem 12
Find the sums. -26+12
View solution Problem 12
Determine each of the values, |14|
View solution Problem 13
Convert the following numbers to standard form. $$ 9.25 \times 10^{2} $$
View solution Problem 13
When simplifying the terms for the following problems, write each so that only positive exponents appear. $$ 4 a^{-6}\left(2 a^{-5}\right) $$
View solution