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Chapter 1: Problem 100

Simplify. $$ \frac{-4-8(-2)}{-9-2(-3)} $$

Short Answer

Expert verified
The simplified expression is -4.

Step by step solution

01

Simplify the Numerator

Start by simplifying the expression in the numerator of the fraction: \(-4 - 8(-2)\). First, multiply \(-8\) by \(-2\) to get \(16\). So, the numerator simplifies to: \(-4 + 16 = 12\).
02

Simplify the Denominator

Now, simplify the expression in the denominator: \(-9 - 2(-3)\). First, multiply \(-2\) by \(-3\) to get \(6\). So, the denominator simplifies to: \(-9 + 6 = -3\).
03

Divide the Numerator by the Denominator

Now that both the numerator and the denominator are simplified, Divide the simplified numerator \(12\) by the simplified denominator \(-3\): \(\frac{12}{-3} = -4.\)
04

State the Final Result

The expression simplifies to:\(-4\.\)

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Key Concepts

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

Numerator Simplification
In rational expressions, the numerator plays a crucial role in the simplification process. The numerator is the top part of a fraction. To simplify it, you need to perform the calculations necessary to reduce it to its simplest form. In our example, the given expression is \(-4 - 8(-2)\). To simplify, you should start by dealing with any multiplication and use the order of operations.
  • Here, multiply \(-8\) and \(-2\). A negative times a negative becomes positive, so the result is \(16\).

  • Substitute this result back into the expression. That makes \(-4 + 16\).

  • Finish simplification by adding \(-4\) and \(16\), which yields \(12\).

It is essential to carefully resolve each operation to avoid mistakes in these simplifications.
Denominator Simplification
After simplifying the numerator, it's time to focus on the denominator. The denominator is the bottom part of a rational expression. It determines how the numerator is divided.In the exercise, the denominator expression is \(-9 - 2(-3)\). Follow these steps to simplify:
  • First, tackle the multiplication process. Multiply \(-2\) by \(-3\), which results in \(6\) because the product of two negatives is positive.

  • Substitute \(6\) back into the expression, turning it into \(-9 + 6\).

  • Combine \(-9\) with \(6\), and you get \(-3\) as the simplified denominator.

Consistently applying the order of operations and carefully working through each step ensures accuracy. A fully simplified denominator sets you up for successful division in the next stage.
Division of Integers
Once you have simplified both the numerator and the denominator, it's time to perform the final step: division. This step involves dividing the simplified numerator by the simplified denominator.In the given expression, after simplification, the numerator is \(12\) and the denominator is \(-3\). Dividing these gives:
  • Perform the division: \(\frac{12}{-3} = -4\).

  • The division of a positive integer by a negative integer results in a negative integer.

Thus, the simplified result of the initial expression is \(-4\). Always double-check both the signs and values involved in the division process. This ensures that the final answer accurately reflects the operations performed.

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Most popular questions from this chapter

Evaluate each expression when \(x=-5, y=4,\) and \(t=10 .\) See Example 6. $$ \frac{9-x}{y+6} $$

Perform the following operations. Write answers in lowest terms. $$ 5+\frac{2}{3} $$

Determine which pairs of actions are commutative. "reading the sports section" and "reading the comics section"

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. The sum of 8 and twice a number is 42

Without calculating, determine whether each answer is positive or negative. Then use a calculator to find the exact difference. \(4.362-7.0086\)
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