Problem 55 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expression \(|\sqrt{2}-5|\) can be rewritten without absolute value bars as \(5 - \sqrt{2}\).

Step 1: Evaluate the expression inside the absolute value brackets

Given the expression, \(|\sqrt{2}-5|\), evaluate \(\sqrt{2}-5\). Since \(\sqrt{2}\) approximates to 1.41, \(1.41 - 5 = -3.59\).

Step 2: Consider the absolute value properties

The absolute value of a number is its distance from zero regardless of direction, which is always positive. So, the absolute value of -3.59 is 3.59. In general terms, for any negative number 'x', the absolute value is '-x'.

Step 3: Write the final answer

So, given this knowledge, we can simplify \(|\sqrt{2}-5|\) to be 3.59 (or 5 - \(\sqrt{2}\) if exact value is required).

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 x 2 equals 4. When dealing with expressions that involve square roots, it's important to understand that these can result in both exact and approximate values. In the given expression, \(|\sqrt{2}-5|\), we encounter \(\sqrt{2}\). The exact value of \(\sqrt{2}\) is an irrational number, approximately equal to 1.41 when rounded to two decimal places. Irrational numbers can't be expressed as a simple fraction and have non-repeating, non-terminating decimal parts. Hence, when we perform operations involving \(\sqrt{2}\), we often use its approximate value to simplify calculations.

Distance from Zero

The concept of distance from zero is crucial to understanding absolute values. Absolute value represents how far a number is from zero on the number line. Whether a number is positive or negative, its absolute value is always non-negative. For instance, both -3 and 3 are 3 units away from zero.

• If the number is positive, the absolute value is the number itself.
• If the number is negative, the absolute value is the positive version of that number.

In our exercise, the expression \(|\sqrt{2}-5|\) simplifies to finding how far the result of \(\sqrt{2} - 5\) is from zero. Calculating this, \(\sqrt{2} - 5 \approx -3.59\), we find that its distance from zero is \(3.59\). This demonstrates absolute value's role in determining a number's magnitude without considering its sign.

Negative Numbers

Negative numbers are values less than zero and are represented by a minus sign \(-\). In mathematics, operations involving negative numbers must be carefully managed to ensure accuracy. When you subtract a larger number from a smaller one, as in \(\sqrt{2} - 5\), the result is negative because 1.41 is less than 5. Recognizing the properties of negative numbers is essential, especially when dealing with absolute values, because absolute values convert any negative result into a positive one. In our context, while \(\sqrt{2} - 5\) gives a negative result of approximately \(-3.59\), applying the absolute value makes it positive, yielding \(3.59\), which reflects the non-negative distance on a number line.

Approximations in Math

Math often requires us to deal with approximations, especially when working with irrational numbers or complex expressions. In problems involving expressions like \(|\sqrt{2}-5|\), exact values can be cumbersome, so approximations make our calculations more manageable. Approximations provide a simplified way to approach an answer, particularly when a calculation involves constants like \(\pi\), \(e\), or square roots.

• Approximations are crucial when an exact value is impractical.
• They simplify complex mathematical operations for easier computation.

In our solution, we used \(\sqrt{2} \approx 1.41\) to easily compute the expression and determine that \(\sqrt{2} - 5 \approx -3.59\). This simplification helps us reach a result that is sufficient for most practical purposes, even if it's not completely precise.