Problem 51 - Algebra 2

Question

Simplify. \(\frac{x-1}{\sqrt{x}-1}\)

Step-by-Step Solution

Answer

The simplified expression is \(\sqrt{x} + 1\).

Step 1: Analyze the Expression

We need to simplify the given expression \( \frac{x-1}{\sqrt{x}-1} \). Notice that the numerator can be written in terms of a familiar algebraic formula.

Step 2: Factor the Numerator

The numerator \(x-1\) can be rewritten as \((\sqrt{x})^2 - 1^2\). This expression is a difference of squares, which factors to \((\sqrt{x} - 1)(\sqrt{x} + 1)\).

Step 3: Cancel Common Factors

The expression is now \( \frac{(\sqrt{x} - 1)(\sqrt{x} + 1)}{\sqrt{x} - 1} \). The common factor \(\sqrt{x} - 1\) in both the numerator and the denominator can be canceled out.

Step 4: Simplify the Expression

After canceling \(\sqrt{x} - 1\), we are left with \(\sqrt{x} + 1\). Therefore, the simplified form of the expression is \(\sqrt{x} + 1\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a powerful tool in algebra, simplifying expressions and solving equations. It involves expressions of the form \( a^2 - b^2 \). This pattern can be factored into \((a-b)(a+b)\).

In our exercise, the expression \(x-1\) is recognized as a difference of squares. Here, \(x\) can be written as \((\sqrt{x})^2\) and \(1\) as \(1^2\). Thus, we have the formula:

• \((\sqrt{x})^2 - 1^2 = (\sqrt{x} - 1)(\sqrt{x} + 1)\)

Understanding this concept allows us to factor expressions effectively, revealing hidden structures that simplify the problem.

Factoring

Factoring is the process of breaking down an expression into simpler "factors" that, when multiplied together, give the original expression.

When we applied the difference of squares rule in our exercise, we factored \(x-1\) into \((\sqrt{x} - 1)(\sqrt{x} + 1)\). These are the simpler components of the expression that are easier to work with.

Factoring, especially using known patterns like difference of squares, helps in reducing complexity. This simplification is crucial because it often exposes opportunities to cancel terms in rational expressions.

Canceling Common Factors

In rational expressions, simplifying often involves canceling common factors. This involves removing identical terms from the numerator and denominator.

After factoring the expression, we observe the common factor \(\sqrt{x} - 1\) in both the numerator and the denominator.

• Expression before canceling: \((\sqrt{x} - 1)(\sqrt{x} + 1)/ (\sqrt{x} - 1)\)
• Once \(\sqrt{x} - 1\) is canceled out, what remains is \(\sqrt{x} + 1\).

By canceling these common terms, we simplify the expression effectively, making it much easier to understand and work with. Remember, cancellation can only occur when the terms are factors of both the numerator and the denominator.