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Chapter 8: Problem 57

Simplify. \(354--\sqrt{3}+5250---\sqrt{3}-416--\sqrt{3}\)

Short Answer

Expert verified
The simplified expression is \(6020 - 3\sqrt{3}\).

Step by step solution

01

Identify and Reorganize Terms

First, identify the integer terms and the square root terms. We have the integers: \(354\), \(5250\), and \(416\). We also have the square root terms \(-\sqrt{3}\), \(-\sqrt{3}\), and \(-\sqrt{3}\). Reorganize the expression: \(354 + 5250 + 416 - \sqrt{3} - \sqrt{3} - \sqrt{3}\).
02

Combine Integer Terms

Add the integer terms together: \(354 + 5250 + 416 = 6020\). So, the expression so far is \(6020 - \sqrt{3} - \sqrt{3} - \sqrt{3}\).
03

Combine Square Root Terms

Add the square root terms together: \(- \sqrt{3} - \sqrt{3} - \sqrt{3} = -3\sqrt{3}\). Now the expression is \(6020 - 3\sqrt{3}\).
04

Write the Final Simplified Expression

The simplified expression is \(6020 - 3\sqrt{3}\).

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Key Concepts

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

Integer Terms
When simplifying expressions, identifying integer terms is a crucial first step. Integer terms are whole numbers without fractions or decimals. In the given expression:
  • 354
  • 5250
  • 416
These numbers should be gathered and then summed. Here, adding these integers yields: \[354 + 5250 + 416 = 6020\] Recognizing and accurately combining integer terms establishes a solid foundation for simplifying more complex expressions.
Square Root Terms
Square root terms involve radicals, often represented by the square root symbol \(\sqrt{}\). In this exercise, the square root terms are all \(-\sqrt{3}\). They are negative to begin with, which affects their combination.
  • Each term is \(-\sqrt{3}\).
  • There are three terms like this in the expression.
To simplify, you add these terms together: \[-\sqrt{3} - \sqrt{3} - \sqrt{3} = -3\sqrt{3}\] Understanding how to handle square root terms is key to simplifying the expression correctly.
Expression Reorganization
Expression reorganization is about rearranging terms for easier computation. In algebraic expressions, clarity is enhanced when similar terms are grouped. For our expression, begin by identifying component parts, then reorder them: \[354 + 5250 + 416 - \sqrt{3} - \sqrt{3} - \sqrt{3}\] This step brings the integers together and the square root terms together, making the next steps鈥攃ombining like terms鈥攎uch more straightforward.
Combining Like Terms
Combining like terms is a fundamental skill for simplifying expressions. This involves merging terms of the same type. In this exercise:
  • Combine integer terms: \(354 + 5250 + 416\) to get \(6020\).
  • Combine square root terms: \(-\sqrt{3} - \sqrt{3} - \sqrt{3}\) to get \(-3\sqrt{3}\).
By following these steps, the expression is simplified to its final form: \[6020 - 3\sqrt{3}\] Effectively combining terms reduces complexity, bringing clarity to the expression.

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