Problem 70 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$|-\sqrt{3}|$$

Step-by-Step Solution

Answer

The absolute value \(|-\sqrt{3}|\) is \sqrt{3}\.

Step 1: Understand Absolute Value

The absolute value of a number is its distance from 0 on the number line, regardless of direction. It is always a non-negative number.

Step 2: Identify the Given Expression

The given expression is \(|-\sqrt{3}|\). The absolute value function is denoted by the vertical bars around the number or expression.

Step 3: Evaluate the Inside of the Absolute Value

Inside the absolute value notation, we have \(-\sqrt{3}\). The square root of 3 (\

Step 4: Find Absolute Value

The absolute value of \(-\sqrt{3}\) is simply \sqrt{3}\ because the absolute value function removes the negative sign.

Key Concepts

Distance from Zero

The absolute value of a number tells us its distance from zero on the number line. This means how many units away from the zero point the number is, regardless of the direction on the number line it lies. For example, the absolute value of both -5 and 5 is 5, because both numbers are 5 units away from zero. The crucial thing to remember here is that absolute value is always a measure of distance, which cannot be negative. Therefore, whenever you calculate an absolute value, you'll end up with a non-negative number.

Non-Negative Number

A non-negative number is any number that is either positive or zero. This means the value is zero or greater. Absolute values are always non-negative because they represent distance. Distance cannot be less than zero. For instance, the absolute value of -3 is 3, and the absolute value of 3 is also 3. Both are non-negative. When working with expressions involving absolute values, like \(|-\frac{1}{2}|\), the result will always be a non-negative number, in this case, \(\frac{1}{2}\).

Square Root

Understanding the square root is essential, especially when it appears within an absolute value expression. The square root of a number is a value which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \[3 \times 3 = 9\]. When dealing with square roots in absolute value expressions, such as \(|-\frac{1}{2}|\), it's important to first understand the square root itself before applying the absolute value rule. In the supplied exercise, \(|-\frac{1}{2}|\) simplifies to \(\frac{1}{2}\) because the square root function and absolute value function effectively make the number non-negative.