Show me how did you expand the second expression

Certainly! Let's expand the expression \( \left( \sqrt{x - 1} \right)^4 \).

First, rewrite the expression using exponent notation:
\[ \left( \sqrt{x - 1} \right)^4 = \left( (x - 1)^{1/2} \right)^4 \]

Using the property of exponents \((a^m)^n = a^{mn}\):
\[ \left( (x - 1)^{1/2} \right)^4 = (x - 1)^{(1/2) \times 4} \]
\[ (x - 1)^{(1/2) \times 4} = (x - 1)^2 \]

Now, expand \((x - 1)^2\) using the binomial theorem or simply by multiplying the binomial by itself:
\[ (x - 1)^2 = (x - 1)(x - 1) \]
\[ (x - 1)(x - 1) = x^2 - x - x + 1 \]
\[ x^2 - x - x + 1 = x^2 - 2x + 1 \]

So, the expanded form of \( \left( \sqrt{x - 1} \right)^4 \) is:
\[ x^2 - 2x + 1 \]

This is a polynomial with terms \( x^2 \), \( -2x \), and \( 1 \), all of which have non-negative integer exponents.

Would you like more details on any specific part or have any questions?

Here are 5 questions you might consider asking next:
1. How can you use the binomial theorem for expanding higher powers?
2. Can you explain the steps for multiplying binomials in detail?
3. How do you determine the degree of a polynomial?
4. What are the common methods for factoring polynomials?
5. How do you graph polynomial functions?

**Tip:** When expanding expressions, carefully apply exponent rules and distribute terms to ensure accuracy.

Explore the detailed expansion of the expression (sqrt(x - 1))^4 into x^2 - 2x + 1 using exponentiation rules and the binomial theorem. This problem covers key concepts in exponents, binomial expansion, and polynomial algebra, suitable for students in Grades 10-12.

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