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Chapter 7: Problem 51

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

Short Answer

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

Step by step solution

01

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.
02

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 \((\sqrt{x} - 1)(\sqrt{x} + 1)\).
03

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.
04

Simplify the Expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

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