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Simplifying expressions means to rewrite them in the simplest form. In this case, we use the Binomial Theorem, which helps us expand expressions like \( (a - b)^2 \). Here, the expression is \( (1 - \sqrt{x})^2 \). Applying the binomial theorem, we write the formula as \( a^2 - 2ab + b^2 \).\
Next, we substitute \( a = 1 \) and \( b = \sqrt{x} \). We calculate \( 1^2 \), \(-2 * 1 * \sqrt{x}\), and \( (\sqrt{x})^2 \). Adding these, we get \( 1 - 2\sqrt{x} + x \).\
This is the simplified form of the expression.

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. Here, we used the binomial theorem, which is a specific form of algebraic manipulation. By substituting specific values, we can break down complex expressions.\
First, recognize the pattern in \( (a - b)^2 \) and identify your \( a \) and \ b \. Then substitute these values into the formula. The steps are: \( a^2 - 2ab + b^2 \).\
Finally, perform calculations for each term individually and then combine them. This allows us to transform \( (1 - \sqrt{x})^2 \) into the expanded form \( 1 - 2\sqrt{x} + x \).\
Through these steps, we can simplify and manipulate algebraic expressions efficiently.

Radicals involve roots, such as square roots. In the expression \( (1 - \sqrt{x})^2 \, \sqrt{x} \) is a radical. Simplifying radicals in an expression can be challenging, but by using algebraic rules, we can manage them effectively.\
When dealing with radicals, remember these points:\

• Square roots can be squared to remove the radical. For example, \( (\sqrt{x})^2 = x \)
• Apply the binomial theorem directly to handle expressions involving radicals. \
  In our example, \(1 - 2\sqrt{x} + x \) shows the result of simplifying \( (1 - \sqrt{x})^2 \), where the radical term \( -2\sqrt{x} \) appears naturally through the binomial expansion.\

These steps help handle radicals within algebraic expressions, making it easier to simplify complex problems.