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Chapter 1: Problem 71

Simplify each expression. See Examples I through 11 . $$ 25-[(3-5)+(14-18)]^{2} $$

Short Answer

Expert verified
The simplified expression is -11.

Step by step solution

01

Simplify Inside the Parentheses

First, simplify the expression inside the parentheses: \( (3-5) + (14-18) \). This results in: 1. \( 3-5 = -2 \)2. \( 14-18 = -4 \)Thus, the expression simplifies to \( -2 + (-4) = -6 \).
02

Handle the Exponent

Next, take the result from Step 1 and apply the square exponent: \( (-6)^2 \). This is calculated as:\((-6) \times (-6) = 36\).
03

Simplify the Whole Expression

Subtract the result from Step 2 from the number outside the brackets: \( 25 - 36 \). Subtracting, you get: \( 25 - 36 = -11 \).

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Key Concepts

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

Simplifying with Order of Operations
To simplify mathematical expressions correctly, it's crucial to follow the order of operations. This means solving parts of an expression using a specific sequence. The traditional way to remember it is through the acronym PEMDAS, which stands for:
  • Parentheses
  • Exponents
  • Multiplication and Division
  • Addition and Subtraction
This order helps us determine which operations to perform first to get an accurate result.
For example, when dealing with the expression \(25 - [(3-5)+(14-18)]^{2}\), we start by addressing what's inside the parentheses, move on to exponents, and finally handle subtraction. Making mistakes with the order can lead to incorrect answers.
Working with Parentheses
Parentheses are used in math to indicate which operations should be performed first. They serve as a way to "group" parts of an expression so we focus on solving what's inside them right from the start.
In the expression \((3-5)+(14-18)\), we first tackle these operations because they are inside the parentheses.
This ensures that any calculations outside the parentheses are based on accurate results.
By dealing with:
  • \(3-5\), we get -2
  • \(14-18\), we get -4
Combining these gives us a simplified value of \(-2 + (-4) = -6\), setting us up for the next steps.
Understanding Exponents
Exponents indicate how many times a value is multiplied by itself. They add an element of repetition and can significantly change the size of a number.
In our example, after simplifying inside the parentheses, we are left with \((-6)\) raised to the power of 2, depicted as \((-6)^2\).
This involves multiplying -6 by itself:
  • \((-6) \times (-6) = 36\)
Remember that a negative number squared results in a positive number because multiplying two negative numbers results in a positive product.
This principle is key for simplifying expressions correctly.
The Role of Subtraction in Expressions
Subtraction is one of the basic operations in mathematics, and its role in simplifying expressions is often towards the final step. After calculating parentheses and exponents, we are ready to perform the subtraction.
Using the previous example, we subtract the squared number from the initial value outside the brackets:
  • \(25 - 36\)
Subtraction here results in -11, which marks the final simplified form of the expression.
While subtraction might seem straightforward, it鈥檚 crucial to handle it at the right step to ensure accuracy in complex scenarios.

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Most popular questions from this chapter

Choose the fraction (s) equivalent to the given fraction. (There may be more than one correct choice.) \(-\frac{1}{7} \quad\) a. \(\frac{-1}{-7}\) b. \(\frac{-1}{7} \quad\) c. \(\frac{1}{-7}\) d. \(\frac{1}{7}\)

Choose the fraction (s) equivalent to the given fraction. (There may be more than one correct choice.) $$ \begin{aligned} &\frac{5}{-(x+y)} \quad \text { a. } \frac{5}{(x+y)} \quad \text { b. } \frac{-5}{(x+y)}\\\ &\frac{-1}{-(x+y)} \cdot \frac{-5}{(x+y)} \end{aligned} $$

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ -(8-t)+(2 t-6) $$

The highest point on land on Earth is the top of Mt. Everest in the Himalayas, at an elevation of \(29,028\) feet above sea level. The lowest point on land is the Dead Sea, between Israel and Jordan, at 1319 feet below sea level. Find the difference in elevations.

Name the only real number that is its own opposite, and explain why this is so.
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