Problem 76 - Contemporary Precalculus

Question

Simplify, and write the given number without using absolute values. $$|\sqrt{2}-2|$$

Step-by-Step Solution

Answer

Question: Simplify the expression $$|\sqrt{2}-2|$$ and write it without using absolute values. Answer: The simplified expression without absolute values is $$2-\sqrt{2}.$$

Step 1: Evaluate the expression inside the absolute value

First, we need to find the value of the expression inside the absolute value: $$\sqrt{2}-2.$$

Step 2: Determine if the result is positive or negative

The next step is to determine if the result of $$\sqrt{2}-2$$ is positive or negative. Since $$\sqrt{2} \approx 1.41$$ and $$1.41 - 2 < 0,$$ the result is negative.

Step 3: Find the absolute value of the negative result

Now that we know the expression within the absolute value is negative, we can find the absolute value of the result by changing the sign from negative to positive. So, the absolute value of $$\sqrt{2}-2$$ is equal to $$2-\sqrt{2}.$$

Step 4: Write the simplified expression without absolute values

Since the value inside the absolute value was negative and we found the positive equivalent, we can now write the expression without absolute values as $$2-\sqrt{2}.$$

Key Concepts

Simplifying Expressions

Simplifying expressions is like solving a puzzle where we aim to make the expression as straightforward as possible. In this context, simplifying involves removing elements such as absolute values or reducing the complexity of mathematical terms.

Here, the expression given is inside absolute value bars: \( |\sqrt{2} - 2| \). To simplify it:

• Evaluate the terms inside the bars first.
• Next, determine whether the result is positive or negative.
• If it's negative, switch the sign to find the absolute value. A negative value inside the absolute bars will become positive outside of it.

For the expression \sqrt{2} - 2, we realize it’s negative since \( \sqrt{2} \ ext{ is about 1.41, and 1.41 - 2 < 0} \). This directs us to adjust the expression to positive form. Simplifying expressions helps reduce them to familiar, easily understandable forms.

Square Roots

Square roots might seem challenging at first, but they simply represent a number that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because 2 multiplied by 2 is 4.

When dealing with expressions like \( \sqrt{2} \), we are looking at a number which, when squared, gives us 2. This value is approximately 1.41.

Square roots are quite common in algebra, and simplifying expressions often requires knowing or estimating the value of these roots. This understanding aids in the comparison or simplification of numerical expressions, especially when dealing with either positive or negative roots.

Negative Numbers

Negative numbers can seem confusing, but they are just numbers less than zero. They represent values below the baseline in mathematics.

In the context of the original exercise, knowing that \( \sqrt{2} - 2 \) produces a negative number \( (approximately -0.59) \) is crucial. When simplifying expressions with absolute value, recognizing if the result inside the bars is negative guides us in modifying it to its positive equivalent.

To handle negative numbers:

• Recognize them as the opposite of positive numbers.
• Understand operations involving negatives, such as subtracting a larger number from a smaller one produces a negative result.
• When inside an absolute value, as in the example, the negative sign is flipped to make the number positive.

Grasping negative numbers facilitates better estimation and manipulation in mathematical problems, as with this solving task.