Problem 100 - Algebra and Trigonometry

Question

Rationalize the numerator. $$ \sqrt{x+1}-\sqrt{x} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression

The given expression is \( \sqrt{x+1} - \sqrt{x} \). The goal is to rationalize the numerator, which means to eliminate the square roots in the numerator by multiplying with a suitable factor.

Step 2: Multiply by Conjugate

To rationalize, multiply the numerator and the denominator by the conjugate of the numerator, which is \( \sqrt{x+1} + \sqrt{x} \). Write this as: \[ \frac{\sqrt{x+1} - \sqrt{x}}{1} \times \frac{\sqrt{x+1} + \sqrt{x}}{\sqrt{x+1} + \sqrt{x}} \]

Step 3: Apply Difference of Squares Formula

The denominator is now \( \sqrt{x+1} + \sqrt{x} \). In the numerator, use the difference of squares formula: \[ (\sqrt{x+1} - \sqrt{x})(\sqrt{x+1} + \sqrt{x}) = (\sqrt{x+1})^2 - (\sqrt{x})^2 \] This simplifies to: \[ (x+1) - x = 1 \]

Step 4: Simplify the Expression

With the numerator simplified to 1, the entire expression now becomes: \[ \frac{1}{\sqrt{x+1} + \sqrt{x}} \]. The expression is rationalized as we have eliminated the square root from the numerator and it only remains in the denominator.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity used to simplify expressions where you have a subtraction between two square terms. It states that:\[ a^2 - b^2 = (a - b)(a + b) \]This means you can rewrite a subtraction between two squares as a product of their sum and difference.

• This is particularly useful in rationalizing expressions because it helps eliminate square roots.
• In our problem, it allows us to turn the product of conjugates into a simple numeric difference.

When applied to roots, like in \[ (\sqrt{x+1})^2 - (\sqrt{x})^2 \],we obtain \[ x + 1 - x = 1 \].This illustrates the power of this identity, making complex expressions much simpler.

Conjugates

The concept of conjugates is central to the process of rationalizing numerators or denominators in expressions involving square roots. A conjugate is formed by changing the sign between two terms. For an expression like \( \sqrt{x+1} - \sqrt{x} \),its conjugate is \( \sqrt{x+1} + \sqrt{x} \).Using conjugates is a clever algebraic trick:

• It relies on the difference of squares to help clear out square roots.
• By multiplying by the conjugate, \( \frac{\sqrt{x+1} - \sqrt{x}}{1} \times \frac{\sqrt{x+1} + \sqrt{x}}{\sqrt{x+1} + \sqrt{x}} \), square roots in the original expression can be rationalized.

Multiplying by conjugates doesn't change the value because you are effectively multiplying by one. Thus, the process keeps the expression equivalent while eliminating unwanted roots.

Simplifying Expressions

Simplifying expressions involves rewriting them in a way that is easier to understand or work with, often making them more elegant or manageable. For the exercise at hand, simplifying involved using the difference of squares and conjugates to clean up the expression.Steps to Simplify:

• Recognize parts of the expression that can be rewritten or have potential for reduction.
• Use identities like the difference of squares to perform algebraic simplifications.
• Keep track of equivalent expressions to ensure changes do not affect the overall value.

In the specific problem, simplifying \[ (\sqrt{x+1} - \sqrt{x})(\sqrt{x+1} + \sqrt{x}) \]yields the clean expression 1, demonstrating how simplification can lead to a neat, more comprehensible form.