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Chapter 3: Problem 78

Clear parentheses and combine like terms. $$11+2(p-8)$$

Short Answer

Expert verified
$$2p - 5$$

Step by step solution

01

Identify terms inside parentheses

The expression given is $$11 + 2(p - 8)$$. Notice that the term inside the parentheses is $$p - 8$$.
02

Distribute the factor outside the parentheses

Distribute the 2 to both terms inside the parentheses: $$2 \times (p - 8) = 2p - 16$$. Now the expression is: $$11 + 2p - 16$$.
03

Combine constants

Combine the constant terms (11 and -16): $$11 - 16 = -5$$. The final expression is: $$2p - 5$$.

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Key Concepts

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

Clearing Parentheses
Understanding how to clear parentheses in an algebraic expression is important. This step helps to simplify the expression by getting rid of any grouped terms. In our exercise, we start with the expression: $$11 + 2(p - 8)$$. To clear the parentheses, we need to distribute the factor outside the parentheses (which is 2 in this case) to each term inside. This process ensures that every term inside the parentheses is accounted for. The expression inside the parentheses here is $$p - 8$$. By distributing, we multiply 2 by each term inside the parentheses:
  • 2 脳 p = 2p
  • 2 脳 (-8) = -16
This clears the parentheses and we now have: $$11 + 2p - 16$$.Next, we move on to combining like terms.
Combining Like Terms
Combining like terms is a key step in simplifying algebraic expressions. Like terms are terms that have the same variable component with the same exponent. In our current expression, $$11 + 2p - 16$$, we have two types of terms: constants (numbers without variables) and variable terms (which include the variable 'p').First, identify the like terms:
  • Constants: 11 and -16
  • Variable terms: 2p (there are no other 'p' terms here to combine with)
Now, combine the constant terms: $$11 - 16 = -5$$. This leaves us with the simplified expression: $$2p - 5$$.The variable term 2p remains as it is because there were no other 'p' terms to combine it with. This step of combining like terms makes the expression much easier to manage.
Distribution in Algebra
The distributive property is a fundamental concept in algebra. It states that a(b + c) = ab + ac. This property allows us to remove parentheses by multiplying the outside factor by each term inside the parentheses.In our example, our original expression was: $$11 + 2(p - 8)$$. Here, we used the distributive property to simplify: $$2(p - 8)$$. This means we performed the following multiplication:
  • 2 脳 p = 2p
  • 2 脳 (-8) = -16
So, the distributed expression became: $$2p - 16$$. By applying the distributive property, we effectively cleared the parentheses and simplified our algebraic expression, making it easier to combine like terms and further simplify.

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