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Chapter 0: Problem 84

Simplify each expression. $$ 3(2 z-4)+8(z-9)+84 $$

Short Answer

Expert verified
\(14z\).

Step by step solution

01

Distribute the numbers

Distribute the numbers outside the parentheses to each term inside the parentheses. For the expression 3(2z - 4) + 8(z - 9) + 84, distribute as follows: \(3 \times 2z - 3 \times 4 + 8 \times z - 8 \times 9 + 84\). This simplifies to: \(6z - 12 + 8z - 72 + 84\).
02

Combine like terms

Combine the like terms from the expression obtained after distribution. Combine the \(z\) terms: \(6z + 8z = 14z\) Combine the constant terms: \(-12 - 72 + 84 = 0\). So the simplified expression becomes: \(14z + 0\), which we can simply write as: \(14z\).

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Key Concepts

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

distributive property
To simplify algebraic expressions, we often start with the distributive property. This property helps us remove parentheses by distributing a factor across the terms inside the parentheses.

For example, in the expression: \(3(2z - 4) + 8(z - 9) + 84\), we need to distribute the numbers 3 and 8 to each term inside their respective parentheses.

Here鈥檚 how it's done step-by-step:
  • First, multiply 3 by each term inside the parentheses: \(3 \times 2z - 3 \times 4\), which gives us \(6z - 12\).
  • Next, do the same with 8 and the terms inside the second set of parentheses: \(8 \times z - 8 \times 9\), which results in \(8z - 72\).
Combining everything, we now have: \(6z - 12 + 8z - 72 + 84\). The distributive property makes it easier to then combine like terms.
combining like terms
After using the distributive property, the next step in simplifying an algebraic expression is combining like terms. Like terms are terms that have the same variable raised to the same power.

In our example, we have simplified it to: \(6z - 12 + 8z - 72 + 84\). Now, we need to combine the like terms.

Here鈥檚 how we do it:
  • Identify terms that are alike. In this case, \(6z\) and \(8z\) are like terms because they both have the variable \(z\).
  • Combine the like terms by adding or subtracting their coefficients: \(6z + 8z = 14z\).
After combining them, our expression becomes \(14z - 12 - 72 + 84\). Now, only constant terms remain to be combined.
constant terms
Finally, we deal with constant terms. Constant terms are simply numbers without any variables attached. In our example, we still need to simplify \(-12 - 72 + 84\).

Here鈥檚 how to combine these constant terms:
  • First, perform the addition or subtraction in sequence: \(-12 - 72 = -84\).
  • Next, add the last constant term: \(-84 + 84 = 0\).
This leaves us with a final simplified expression of: \(14z\).
Notice that when constant terms add up to zero, they can be omitted from the final expression.

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Most popular questions from this chapter

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. Determine the total expenditure for other health-related out-of-pocket expenses for the year \(2005 .\)

Determine the degree of the polynomial. $$ -8 p^{2} q r^{5}+4 p q^{8} r^{2}+5 p^{3} q^{3} r $$

Perform the indicated operations and simplify. $$ (p-2)^{3} $$

Write the polynomial in descending order. Then identify the leading coefficient and degree of the polynomial. $$ \frac{4}{5} c^{2}+c^{5} $$

We know that \((a+b)^{2}=a^{2}+2 a b+b^{2}\). Derive a special product formula for \((a+b)^{3}\).
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