Problem 71 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(13 p+4(4-8 p)\)

Step-by-Step Solution

Answer

-19p + 16

Step 1: Distribute the 4

First, distribute the 4 in the expression inside the parentheses: \[ 4(4 - 8p) = 4 \times 4 - 4 \times 8p \] This simplifies to:\[ 16 - 32p \]So the expression now is:\[ 13p + 16 - 32p \]

Step 2: Combine like terms

Combine the like terms (the terms that have \( p \) in them):\[ 13p - 32p + 16 \]Simplify the terms with \( p \):\[ (13 - 32)p + 16 \]Which simplifies to:\[ -19p + 16 \]

Key Concepts

Distributive Property

The distributive property is a fundamental idea in algebra. It allows us to simplify expressions where a number or variable is multiplied by a group of numbers inside parentheses. In our exercise, we start with the expression:
\(13p + 4(4 - 8p)\)
To apply the distributive property, we multiply 4 by each term inside the parentheses:
\[4(4 - 8p) = 4 \times 4 - 4 \times 8p\]
Breaking it down:

• First, multiply 4 by 4, giving us 16.
• Next, multiply 4 by -8p, which gives us -32p.

After distribution, we have:\(16 - 32p\)
So, replacing the original part of the expression, we get:
\(13p + 16 - 32p\).
By using the distributive property, we've transformed a more complex expression into a simpler one.

Combining Like Terms

Combining like terms is a technique used to simplify algebraic expressions. Like terms are terms that have the same variable raised to the same power. In our example, the resulting expression from the distributive step was:
\(13p + 16 - 32p\)
We identify the like terms involving the variable 'p':
\(13p\) and \(-32p\).
To combine these terms, we simply add or subtract their coefficients (the numbers in front of the variables):
\(13p - 32p\)
This results in:
\((13 - 32)p = -19p\)
So, after combining like terms, our expression now looks like this:
\(-19p + 16\)
Combining like terms makes our algebraic expression simpler and easier to work with.

Intermediate Algebra

Intermediate algebra encompasses concepts that extend beyond the basics. It includes skills like using the distributive property, combining like terms, and solving equations. These are essential tools for solving more complex problems.
Let's put it all together with our given exercise. We had to simplify:
\(13p + 4(4 - 8p)\).
By first distributing the 4, we turned it into:
\(13p + 16 - 32p\).
Then, we combined like terms to get:
\(-19p + 16\).
These steps showcase fundamental techniques of intermediate algebra.
Understanding these concepts deeply helps in tackling various algebra exercises with confidence and precision.

• The distributive property helps to break down expressions.
• Combining like terms helps to simplify and reveal a clearer form.

Master these techniques to excel in your algebra journey.