\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} please solve this problems

To solve the expression \(\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\), we need to simplify it step by step. Let's break it down for clarity.

Step 1: Understanding the Components

The expression consists of a square root of a fraction. The numerator is \(1 - \sqrt{x}\) and the denominator is \(1 + \sqrt{x}\). Before we proceed, it's important to note that the square root function is defined only for non-negative values. Therefore, we need to ensure that the expression inside the square root is non-negative.

Step 2: Analyzing the Fraction

We can analyze the fraction \(\frac{1 - \sqrt{x}}{1 + \sqrt{x}}\). For this fraction to be non-negative, both the numerator and denominator must either be both positive or both negative.

• Numerator: \(1 - \sqrt{x} \geq 0\) implies \(\sqrt{x} \leq 1\), which means \(x \leq 1\).
• Denominator: \(1 + \sqrt{x} > 0\) is always true for \(x \geq 0\) since \(\sqrt{x}\) is non-negative.

Thus, the condition for the entire expression to be valid is \(0 \leq x \leq 1\).

Step 3: Simplifying the Expression

Now, let's simplify the expression itself. We can rewrite the square root of the fraction as follows:

\(\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} = \frac{\sqrt{1 - \sqrt{x}}}{\sqrt{1 + \sqrt{x}}}\)

Step 4: Further Simplification

Next, we can simplify the square roots in the numerator and denominator. However, without specific values for \(x\), we can’t simplify further in a general sense. Instead, we can evaluate the expression for specific values of \(x\) within the defined range.

Example Evaluations

• If \(x = 0\):

  \(\sqrt{\frac{1 - \sqrt{0}}{1 + \sqrt{0}} = \sqrt{\frac{1 - 0}{1 + 0}} = \sqrt{1} = 1\)

• If \(x = 1\):

  \(\sqrt{\frac{1 - \sqrt{1}}{1 + \sqrt{1}} = \sqrt{\frac{1 - 1}{1 + 1}} = \sqrt{0} = 0\)

• If \(x = 0.25\):

  \(\sqrt{\frac{1 - \sqrt{0.25}}{1 + \sqrt{0.25}} = \sqrt{\frac{1 - 0.5}{1 + 0.5}} = \sqrt{\frac{0.5}{1.5}} = \sqrt{\frac{1}{3}} \approx 0.577\)

Final Thoughts

The expression \(\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}}\) can be evaluated for any \(x\) in the range \(0 \leq x \leq 1\). The key steps involve ensuring the expression under the square root is non-negative and simplifying the fraction appropriately. This approach not only helps in solving the problem but also reinforces the importance of understanding the conditions under which mathematical expressions are defined.