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Chapter 7: Problem 135

In the following exercises, simplify using the distributive property. $$9(5 u+8)+2(u-6)$$

Short Answer

Expert verified
47u + 60

Step by step solution

01

Apply the Distributive Property to the First Term

Distribute the 9 to both terms inside the parentheses: \[9(5u + 8) = 9 \times 5u + 9 \times 8\Rightarrow 45u + 72\]
02

Apply the Distributive Property to the Second Term

Distribute the 2 to both terms inside the parentheses: \[2(u - 6) = 2 \times u + 2 \times (-6)\Rightarrow 2u - 12\]
03

Combine Like Terms

Combine the like terms from both distributed expressions:\[45u + 72 + 2u - 12\Rightarrow 45u + 2u + 72 - 12\Rightarrow 47u + 60\]

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is about reducing them to their simplest form. This usually means using mathematical properties and combining like terms to make the expression more manageable. In the exercise given, we start by using the distributive property. This property helps us remove parentheses. For example, distributing 9 into \(5u + 8\): \[ 9 \times (5u + 8) = 45u + 72 \]. Similarly, distribute 2 into \(u - 6\): \[ 2 \times (u - 6) = 2u - 12 \]. After distribution, we combine like terms to further simplify.
Combining Like Terms
Combining like terms means grouping terms with the same variables and exponents. This makes the expression easier to work with. In our exercise, after using the distributive property, we get: \[45u + 72 + 2u - 12\]. Here, \(45u\) and \(2u\) are like terms because they both have the variable \(u\). We combine them: \[ 45u + 2u = 47u \]. Similarly, \(72\) and \(-12\) are constants and can be combined: \[ 72 - 12 = 60 \]. This results in the simplified expression: \[ 47u + 60 \].
Polynomial Operations
Understanding polynomial operations is essential. A polynomial is an expression made up of terms added or subtracted together, often involving variables raised to whole number powers. Operations on polynomials include addition, subtraction, and distribution. For instance, the initial expression \( 9(5u + 8) + 2(u - 6) \) is a combination of polynomial terms. We distribute first, as shown: \( 45u + 72 \) and \( 2u - 12 \). Then, we add the polynomials by combining like terms: \[ 45u + 2u + 72 - 12 = 47u + 60 \]. Polynomial operations often make larger or more complex expressions manageable and solvable.

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