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Chapter 1: Problem 50

Use the order of operations to simplify. $$-7(4)+(-8+6)^{4}+5$$

Short Answer

Expert verified
The simplified expression using the order of operations for \(-7(4)+(-8+6)^{4}+5\) is \(-7\).

Step by step solution

01

Perform operations inside parentheses

We start by simplifying the expression inside the parentheses. \(-7(4)+(-8+6)^{4}+5 = -7(4)+(-2)^{4}+5\)
02

Evaluate exponents

Now, we calculate the exponent. \(-7(4)+(-2)^{4}+5 = -7(4)+16+5\)
03

Perform multiplication

Next, we perform the multiplication. \(-7(4)+16+5 = -28+16+5\)
04

Perform addition and subtraction

Finally, we perform addition and subtraction from left to right. \(-28+16+5 = -12+5 = -7\) So the simplified expression is: \(-7\).

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Key Concepts

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

Simplifying Expressions
When dealing with expressions like -7(4) + (-8+6)^4 + 5, it's important to simplify step by step. Simplifying an expression means making it as straightforward as possible. We break down complex expressions into simpler parts. More specifically, we evaluate and eliminate parentheses, exponents, and other operations.
This process often involves following a specific order of operations to ensure accuracy. The main goal of simplifying is to make the expression easier to understand and solve. For instance, in this expression, focusing on smaller sections like -7(4) or (-8+6) can help simplify the larger puzzle. Remember, each step is crucial in reaching the simplest form.
Parentheses
Parentheses signal which part of an expression should be dealt with first. In the given expression, we find parentheses around -8+6, indicating it requires initial attention.
By simplifying what's inside the parentheses first, we can manage the expression more easily. Here, the parentheses contain (-8+6), which is straightforward:
  • -8+6 simplifies to -2.
After simplifying within the parentheses, the equation begins to take on a less daunting form. Handling parentheses first is vital in maintaining the right order of operations, ensuring everything fits together accurately.
Exponents
Once the expression inside the parentheses is simplified, we turn to exponents. An exponent indicates how many times a number, the base, is multiplied by itself. In our expression, (-2) is raised to the power of 4: (-2)^4.
Here, this means:
  • -2 multiplied by itself four times.
  • This results in 16 (-2 脳 -2 脳 -2 脳 -2 = 16).
Exponents can change the value of a number significantly, making this step essential. Accurately evaluating exponents helps to correctly alter the expression's components.
Addition and Subtraction
The final step involves addition and subtraction, primarily handled from left to right. Once the expression has been simplified by clearing parentheses and evaluating exponents, we often face simple arithmetic. In this expression, we perform: -28 + 16 + 5.
Working through this, we find:
  • -28 plus 16 equals -12.
  • Then, -12 plus 5 equals -7.
Following these sequence rules ensures that each number is accurately worked through, reflecting changes brought by addition and subtraction. Always remind yourself of the importance of moving from left to right, especially in these final steps.

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Most popular questions from this chapter

Write a mathematical expression for each and simplify. 18 less than the sum of \(-5\) and 11.

Write a mathematical expression for each and simplify. The product of \(-12\) and 6

Use a signed number to represent the change in each situation. In Michael Jordan's last season with the Chicago Bulls \((1997-1998)\) the average attendance at the United Center was \(23,988 .\) During the \(2002-2003\) season, the average attendance fell to \(19,617 .\) This was a decrease of 4371 people per game. (Total Basketball: The Ultimate Basketball Encyclopedia, by Ken Shouler, Bob Ryan, Sam Smith, Leonard Koppett, and Bob Bellotti, \(\oplus\) 2003, SPORT Media Pub, Inc.)

Rewrite each expression using the distributive property. Simplify if possible. $$-3(h-9)$$

Rewrite each expression using the distributive property. Simplify if possible. $$10(m+5 n-3)$$
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