Problem 19 - College Algebra

Question

rewrite each expression without absolute value bars. $$ |\sqrt{2}-5| $$

Step-by-Step Solution

Answer

The expression \( |\sqrt{2}-5| \) without absolute value bars is \(5 - \sqrt{2}\).

Step 1: Establish the sign of the expression

We need to determine whether \(\sqrt{2} - 5\) is positive or negative. Since \(\sqrt{2} < 2 < 5\), we can say that \(\sqrt{2} - 5\) is a negative number.

Step 2: Rewrite the absolute value expression without the bars

Remember that the absolute value of a negative number is its opposite, which is a positive number. Hence, \( |\sqrt{2} - 5| \) simplifies to \(5 - \sqrt{2}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is essentially about making them easier to understand and work with. This can involve a variety of techniques, depending on what type of expression you are dealing with.
In the case of absolute value expressions, simplifying is often about understanding and using the properties of absolute values. An absolute value represents the distance of a number from zero on a number line, so it is always a non-negative number. When you come across an expression like \[|\sqrt{2} - 5|\]The aim is to rewrite it in a simpler form without the absolute value bars by evaluating whether the expression inside is positive or negative.
To simplify:

• Determine the sign of the expression inside the bars.
• If the expression is negative, take the opposite.
• If it's positive or zero, the expression remains the same without the bars.

This process ensures you get a simplified, clearer formulation that is easier to work with in further calculations or problem-solving.

Positive and Negative Numbers

Positive and negative numbers are fundamental in mathematics, representing values greater than zero and less than zero, respectively. Understanding their properties is crucial when working with expressions and particularly when dealing with absolute values.In the context of the original exercise, the number inside the absolute value expression \[\sqrt{2} - 5\]needs to be evaluated for its sign to simplify the expression appropriately.
Here’s how you approach this:

• Calculate or estimate the value of the numbers involved. \(\sqrt{2}\) is approximately 1.41.
• Compare the numbers: \(\sqrt{2} < 5\), so \(\sqrt{2} - 5\) is negative, approximately \(-3.59\).
• Since we know it is negative, the absolute value transformation converts it to its "opposite", resulting in a positive value.

Recognizing whether each part of your expression is positive or negative sets the foundation for further mathematics involving sums, differences, products, or even functions like absolute values.

Square Roots

Square roots are numbers that produce a specified quantity when multiplied by itself. Square roots are not always integers and can often be irrational numbers, like \(\sqrt{2}\). Understanding and working with square roots is essential in simplifying and evaluating expressions.In exercises involving square roots:

• Recognize that square roots of non-perfect squares will be irrational, e.g., \(\sqrt{2} \approx 1.41\).
• Remember that you can approximate these values when necessary to make comparison or arithmetic easier. In the case of \(\sqrt{2} - 5\), approximating helps to quickly determine that \(\sqrt{2} < 5\) and thereby conclude the expression is negative.
• Apply the property that the square root function returns non-negative values, which is important when determining the sign of components within an absolute value expression.

Working comfortably with square roots allows you to assess the value of expressions quickly and accurately, facilitating more advanced mathematical operations.