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Chapter 1: Problem 3

Write the given quantity without the absolute value symbols. $$ |8-\sqrt{63}| $$

Short Answer

Expert verified
The expression without absolute value symbols is \( 8 - \sqrt{63} \).

Step by step solution

01

Evaluate the expression inside the absolute value

First, calculate the value of \( \sqrt{63} \). Since \( \sqrt{63} \approx 7.94 \), compare it to 8. So, the expression \( 8 - \sqrt{63} \approx 8 - 7.94 = 0.06 \).
02

Determine the sign of the expression

Since \( 8 - \sqrt{63} = 0.06 \) is positive, it means that the value inside the absolute value is greater than zero.
03

Remove the absolute value symbols

When the value inside the absolute value symbols is positive, the absolute value does not change the expression. Therefore, \( |8 - \sqrt{63}| = 8 - \sqrt{63} \).

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Key Concepts

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

Evaluate Expression
In mathematical problems, evaluating an expression is a critical skill. It means to compute or find the value of a given mathematical expression. When you evaluate an expression step by step, you simplify it to the most reduced and understandable form.
  • Start with computing any operations inside parentheses or symbols like square roots, which are prioritied in order of operations.
  • Then, move onto subtraction, addition, multiplication, or division as required.
In our problem, we had the expression \(8 - \sqrt{63}\). The task was to evaluate \(\sqrt{63}\) first.
This means approximating the square root of 63 to the nearest decimal, which is approximately 7.94.
Remove Absolute Value
The concept of absolute value involves the distance of a number from zero on a number line, making it always non-negative. Absolute values are represented with vertical bars \(|...|\),and when you encounter these symbols in an expression you may need to simplify by removing them.
There are two main guidelines when removing absolute value symbols:
  • If the expression inside the absolute value is positive, you can simply remove the bars and keep the expression as it is.
  • If the expression is negative, remove the bars and reverse the sign of the expression inside.
In this particular exercise, since the evaluated expression \(8 - \sqrt{63} \approx 0.06\) turned out to be positive, we were able to remove the absolute value signs directly.
Positive Expression
A positive expression in mathematics is any expression or number that is greater than zero. Understanding whether an expression is positive or negative is important, especially when dealing with absolute values.
To determine if an expression is positive:
  • Evaluate the components individually using standard arithmetic.
  • Combine the evaluated parts in the order dictated by operations.
If the result is greater than zero, the expression is positive.
For our instance with \(8 - \sqrt{63} = 0.06\), since 0.06 is greater than zero, the expression within the absolute value was positive, allowing for straightforward removal of the absolute value symbols.

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