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Chapter 14: Problem 96

Simplify. $$ \left(a_{1}+8 d\right)-\left(a_{1}+7 d\right) $$

Short Answer

Expert verified
d

Step by step solution

01

- Distribute the Negative Sign

The expression provided is \(\big(a_{1}+8d\big)-\big(a_{1}+7d\big)\). Start by distributing the negative sign inside the parenthesis: \(a_{1} + 8d - a_{1} - 7d\).
02

- Combine Like Terms

Next, combine the terms that are alike. Combine the \(a_{1}\) terms and the \(d\) terms: \(a_{1} - a_{1} + 8d - 7d\).
03

- Simplify the Expression

Now simplify by calculating within each term group. Since \(a_{1} - a_{1} = 0\) and \(8d - 7d = d\), the simplified expression is \d\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra. It states that a term outside a parenthesis can be multiplied with each term inside the parenthesis. This property is useful for simplifying expressions. For example, in the expression \((a_1 + 8d) - (a_1 + 7d)\)\, you distribute the negative sign: \((a_1 + 8d) - a_1 - 7d\). This step helps to eliminate the parentheses and makes it easier to combine like terms later on.
Combining Like Terms
Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. In the simplified expression, \(a_1 + 8d - a_1 - 7d\), you see that \a_1\ and \a_1\ are like terms, as are \8d\ and \7d\. You can combine them together. \a_1 - a_1\ gives you \0\, and \8d - 7d\ simplifies to \d\. This leaves you with a much simpler expression: \d\.
Algebra
Algebra involves working with variables and constants to solve equations and simplify expressions. In this exercise, you worked with algebraic expressions and applied basic algebra rules. By distributing the negative sign and combining like terms, you simplified the expression from \((a_1 + 8d) - (a_1 + 7d)\) to \d\. Understanding these fundamental concepts in algebra makes solving complex equations easier.

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