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Chapter 4: Problem 620

In the following exercises, simplify. $$\frac{8 \cdot 7+5(8-10)}{9 \cdot 3-6 \cdot 4}$$

Short Answer

Expert verified
The simplified expression is \( \frac{46}{3} \).

Step by step solution

01

Simplify the numerator

First, calculate inside the parentheses and perform multiplications: For the numerator: 1. Simplify inside the parentheses: \[ 8 \times 7 + 5(8-10) = 8 \times 7 + 5 \times (-2) \] 2. Perform the multiplications: \[ 8 \times 7 = 56 \] \[ 5 \times (-2) = -10 \] 3. Add the results together: \[ 56 + (-10) = 46 \] So, the simplified numerator is 46.
02

Simplify the denominator

Now, calculate the denominator: 1. Perform the multiplications: \[ 9 \times 3 = 27 \] \[ 6 \times 4 = 24 \] 2. Subtract the results: \[ 27 - 24 = 3 \] So, the simplified denominator is 3.
03

Divide the numerator by the denominator

Finally, divide the simplified numerator by the simplified denominator: \[ \frac{46}{3} \] So, the simplified expression is \( \frac{46}{3} \).

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Key Concepts

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

Order of Operations
Understanding the order of operations is crucial in simplifying algebraic expressions. This rule dictates the sequence in which mathematical operations should be performed to ensure accurate results.
The standard order can be remembered using the acronym PEMDAS, which stands for:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In our problem, we started by simplifying the expression inside the parentheses, which is the first step according to PEMDAS. Then, we proceeded with multiplication before finally performing addition. By following these steps, we achieve the correct simplified result.
Numerator and Denominator
For fractions, understanding the parts is essential.
The numerator is the top part of the fraction, while the denominator is the bottom part. They represent how many parts out of the whole you are dealing with.
In our exercise, once we have simplified the numerator and denominator separately, we combine them in the form of a fraction.
Specifically, the numerator was simplified by handling operations inside the parentheses and then multiplying. The resulting simplified numerator was 46.
For the denominator, we performed multiplication and then subtraction, leading to a simplified denominator of 3.
Arithmetic Operations
Arithmetic operations involve basic mathematical processes: addition, subtraction, multiplication, and division. These are foundational to solving algebraic problems.
Let's break down their role in our exercise:
  • Multiplication: We performed multiplication early on both the numerator and the denominator. For example, we multiplied 8 and 7, and 5 with -2 in the numerator.
  • Addition/Subtraction: Addition and subtraction were used after multiplication. In the numerator, -10 was added to 56, which is essentially a subtraction.
  • Division: Finally, division was used to simplify the fraction, dividing the simplified numerator (46) by the simplified denominator (3).
These operations allow us to simplify complex expressions into their simplest form step by step.

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Most popular questions from this chapter

In the following exercises, simplify. $$ \frac{2^{3}-2^{2}}{\left(\frac{3}{4}\right)^{2}} $$

In the following exercises, solve the equation. $$ \frac{x}{5}=-10 $$

In the following exercises, change to equivalent fractions using the given LCD. $$\frac{1}{3} \text { and } \frac{1}{5}, \quad L C D=15$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x+\left(-\frac{11}{12}\right)\) when (a) \(x=\frac{11}{12} \quad\) (b) \(x=\frac{3}{4}\)

In the following exercises, translate and solve. Three-eighths of \(y\) is 24
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