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Chapter 1: Problem 25

evaluate each of the given expressions by performing the indicated operations. $$8-3(-4)$$

Short Answer

Expert verified
The evaluated expression is 20.

Step by step solution

01

Identify the Components

You are given the expression \(8 - 3(-4)\). Here, \(3(-4)\) is a multiplication operation within the expression.
02

Perform Multiplication

Calculate the result of the multiplication. Multiply \(3\) by \(-4\), which gives you \(-12\). Replace \(3(-4)\) with \(-12\) in the expression.
03

Simplify the Expression

Now simplify the expression \(8 - (-12)\). Subtracting a negative number is the same as adding its positive, so \(8 - (-12)\) becomes \(8 + 12\).
04

Perform the Addition

Finally, add \(8\) and \(12\) to get \(20\).

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Key Concepts

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

Multiplication of Integers
When it comes to multiplying integers, it's essential to remember the rules regarding their signs. Here鈥檚 a quick guide to help you:
  • Positive 脳 Positive: The result is always positive. For example, multiplying 4 by 5 gives 20.
  • Positive 脳 Negative: The result is always negative. Think of it as owing money; if you have 3 bills, each worth -4 dollars, you owe 12 dollars. So, 3 脳 (-4) = -12.
  • Negative 脳 Negative: This surprising one yields a positive. Picture negative feedback removing negativity in a conversation. So, (-3) 脳 (-4) = 12.
Multiplying integers gets easier once you know these rules by heart, reducing errors during calculations and making it more intuitive.
Simplifying Expressions
Simplifying expressions means reorganizing them to be more straightforward without changing their value. It involves:
  • Combining like terms. These terms have identical variables raised to the same power. For instance, in 2x + 3x, combine to get 5x.
  • Performing operations where possible. In our original exercise, that means executing the multiplication first, then adjusting the expression accordingly.
  • Reducing complexity. Simplification often involves removing unnecessary brackets and simplifying the overall look of the expression. For example, when you see an expression like 8 - (鈥�12), it's easy to convert into 8 + 12 by understanding the rules of negative numbers.
The goal is an expression that鈥檚 as simple as possible, making further algebraic manipulations more manageable.
Negative Numbers
Negative numbers might appear confusing, but mastering them is crucial in algebra. Here's how they work in different operations:
  • Addition with Negatives: Adding a negative number is identical to subtraction. For instance, 8 + (-3) becomes 8 - 3, equals 5.
  • Subtraction with Negatives: Subtracting a negative converts to addition. So, 8 - (-3) turns into 8 + 3, which equals 11. It's similar to reversing a direction twice, ending back where you started.
  • Multiplication with Negatives: Recall, multiplying a positive with a negative outputs a negative. Thus, 3 脳 (-4) leads to -12, as explored in the original problem.
Understanding how negative numbers function with these operations ensures precision in calculations and a stronger grasp of algebraic concepts.

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Most popular questions from this chapter

Perform the indicated calculations by first expressing all numbers in scientific notation.In a microwave receiver circuit, the resistance \(R\) of a wire \(1 \mathrm{m}\) long is given by \(R=k / d^{2}\), where \(d\) is the diameter of the wire. Find \(R\) if \(k=0.00000002196 \Omega \cdot \mathrm{m}^{2}\) and \(d=0.00007998 \mathrm{m}\).

Solve the given problems. A car costs \(\$ 38,000\) new and is worth \(\$ 24,000\) 2 years later. The annual rate of depreciation is found by evaluating \(100(1-\sqrt{V / C}),\) where \(C\) is the cost and \(V\) is the value after 2 years. At what rate did the car depreciate? (100 and 1 are exact.)

By multiplication, show that \((x+y)\left(x^{2}-x y+y^{2}\right)=x^{3}+y^{3}.\)

Answer the given questions. Is the following simplification correct? Explain. $$\begin{aligned} 2 x-3 y+5-(4 x-y+3) &=2 x-3 y+5-4 x-y+3 \\ &=-2 x-4 y+8 \end{aligned}$$

Simplify the given algebraic expressions. Each of two suppliers has \(2 n+1\) bundles of shingles costing \(\$ 30\) each and \(n-2\) bundles costing \(\$ 20\) each. How much more is the total value of the \(\$ 30\) bundles than the \(\$ 20\) bundles?
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