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Chapter 4: Problem 52

Simplify. $$(3 a)(-2)$$

Short Answer

Expert verified
The simplified form of \((3a)(-2)\) is \(-6a\).

Step by step solution

01

Identify the terms to be multiplied

The terms to be multiplied are \(3a\) and \(-2\). The term \(-2\) is essentially a coefficient that we will multiply with \(3a\).
02

Apply the multiplication rule

The rule of multiplication states that multiplying a term by -2 gives a negated form of that term. So, we multiply \(3a\) by \(-2\).
03

Simplify the expression

Let's simplify the expression by performing the multiplication: \((3a)(-2)\) simplifies to \(-6a\). This is the result we were seeking.

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Key Concepts

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

Multiplication of Terms
When we talk about multiplying terms in algebra, we are essentially dealing with two components: the coefficients (numbers) and the variables (letters). In the example given, we have terms like \(3a\) and \(-2\). Here:
  • 3 is the coefficient of the variable \(a\).
  • -2 is a standalone coefficient.
To multiply \(3a\) by \(-2\), first multiply the coefficients. So you perform \(3 \times -2\).
This results in -6. Next, attach the variable \(a\) to get the final result, which is \(-6a\).
When you multiply coefficients, remember the sign rules:
  • Positive times positive is positive.
  • Negative times positive (or vice versa) is negative.
  • Negative times negative is positive.
Simplifying Expressions
Simplifying expressions involves combining like terms and reducing an expression to its simplest form. The main objective is to make the expression as straightforward as possible. Let's consider the expression \( (3a)(-2)\).
To simplify, you perform the multiplication: \(3 \times -2 = -6\). You then apply this coefficient to the variable \(a\), resulting in \(-6a\).
Sometimes, this may involve distributing terms, factoring, or combining like terms in more complex expressions.

Key steps for simplifying:
  • Combine like terms: Variables and exponents must match.
  • Remember your order of operations (PEMDAS/BODMAS).
  • Always perform multiplication and division from left to right first.
Simplifying expressions helps in solving equations and inequalities with ease.
Negative Coefficients
Negative coefficients can initially seem confusing, but they just follow regular multiplication and addition rules. A coefficient is a number placed in front of a variable, like the \(-2\) in the expression \( (3a)(-2)\).
In this case, you multiply the negative coefficient by the positive coefficient to create \(-6a\).
When dealing with negative coefficients, remember:
  • Negative times positive yields a negative result.
  • Always keep track of the signs and ensure you apply them correctly in operations.
  • When simplifying, treat negative coefficients like expense or loss; they 'subtract' from the outcome.
Negative coefficients are pivotal in understanding balance, such as when solving for \(x\) in equations.
They are not to be avoided but understood, as they offer depth and meaning in expressions and calculations.

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Most popular questions from this chapter

The chemical formula for glucose (sugar) is \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} .\) This formula means that there are 12 hydrogen atoms for every 6 carbon atoms and every 6 oxygen atoms in each molecule of glucose (see the figure at the right). If \(x\) represents the number of oxygen atoms in a pound of sugar, express the number of hydrogen atoms in the pound of sugar in terms of the number of oxygen atoms.

Translate into a variable expression. Then simplify. a number minus the sum of the number and ten

Simplify. $$8-(12+4 y)$$

Simplify. $$-8\left(3 y^{2}-12\right)$$

Simplify. $$3(a-b)-(a+b)$$
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