Problem 71
Question
Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{x}+1}{\sqrt{x}-1}\)
Step-by-Step Solution
Verified Answer
\(\frac{x-1}{x - 2\sqrt{x} + 1}\) is the rationalized form.
1Step 1: Identify the expression to rationalize
We want to rationalize the numerator of the fraction \(\frac{\sqrt{x}+1}{\sqrt{x}-1}\). This means we should eliminate any radical in the numerator.
2Step 2: Multiply numerator and denominator by the conjugate of the numerator
The conjugate of \(\sqrt{x} + 1\) is \(\sqrt{x} - 1\). To rationalize the numerator, multiply both the numerator and the denominator of the fraction by this conjugate:\[\frac{\sqrt{x}+1}{\sqrt{x}-1} \cdot \frac{\sqrt{x}-1}{\sqrt{x}-1} = \frac{(\sqrt{x}+1)(\sqrt{x}-1)}{(\sqrt{x}-1)(\sqrt{x}-1)}\]
3Step 3: Simplify the new numerator
Use the difference of squares formula, \( (a+b)(a-b) = a^2 - b^2 \), to simplify the new numerator:\[(\sqrt{x}+1)(\sqrt{x}-1) = (\sqrt{x})^2 - 1^2 = x - 1\]
4Step 4: Simplify the denominator
Calculate the square of the original denominator:\[(\sqrt{x}-1)^2 = (\sqrt{x})^2 - 2\sqrt{x} + 1 = x - 2\sqrt{x} + 1\]
5Step 5: Write the final rationalized expression
The expression is now simplified to have a rationalized numerator:\[\frac{x-1}{x - 2\sqrt{x} + 1}\]
Key Concepts
RadicalsConjugatesDifference of Squares FormulaSimplifying Expressions
Radicals
Radicals are expressions that contain a root sign, such as a square root (\(\sqrt{}\)), cube root, or higher. In math, particularly algebra, radicals can sometimes complicate expressions. For instance, the expression \(\sqrt{x} + 1\) contains a radical. Since radicals are irrational numbers, they are less straightforward to work with than whole numbers or simple fractions.
To simplify expressions, especially fractions, removing radicals from the numerator or denominator can make calculations easier. This process is known as rationalization. When rationalizing a numerator, we aim to eliminate these radical parts from the top of a fraction. This is often done using techniques like conjugates, which we'll dive into next.
To simplify expressions, especially fractions, removing radicals from the numerator or denominator can make calculations easier. This process is known as rationalization. When rationalizing a numerator, we aim to eliminate these radical parts from the top of a fraction. This is often done using techniques like conjugates, which we'll dive into next.
Conjugates
The "conjugate" is a nifty trick used to remove radicals from expressions. Specifically, the conjugate of a binomial like \( a + b \) is \( a - b \). For our exercise, the numerator \( \sqrt{x} + 1 \) has a conjugate \( \sqrt{x} - 1 \).
Why do conjugates work? When we multiply a binomial by its conjugate, the radicals cancel out through a special pattern known as the "difference of squares." This makes it a powerful tool for rationalizing either numerators or denominators.
Here's the process:
Why do conjugates work? When we multiply a binomial by its conjugate, the radicals cancel out through a special pattern known as the "difference of squares." This makes it a powerful tool for rationalizing either numerators or denominators.
Here's the process:
- Identify the numerator (or denominator) you want to rationalize.
- Find its conjugate by changing the sign of the second term.
- Multiply both the numerator and the denominator by this conjugate.
Difference of Squares Formula
The difference of squares formula is an algebraic identity that states: \( (a+b)(a-b) = a^2 - b^2 \). This rule allows us to simplify expressions formed by binomials and their conjugates.
In our exercise, we leveraged this formula when multiplying \((\sqrt{x} + 1)(\sqrt{x} - 1)\). Applying the formula, the terms \(a\) and \(b\) correspond to \(\sqrt{x}\) and 1 respectively, forming the result \(x - 1\).
The beauty of the difference of squares is in its ability to eliminate the middle terms and, notably, the radicals. It's a go-to technique when you encounter binomials multiplied by their conjugates. It simplifies the expression quickly and keeps your work organized.
In our exercise, we leveraged this formula when multiplying \((\sqrt{x} + 1)(\sqrt{x} - 1)\). Applying the formula, the terms \(a\) and \(b\) correspond to \(\sqrt{x}\) and 1 respectively, forming the result \(x - 1\).
The beauty of the difference of squares is in its ability to eliminate the middle terms and, notably, the radicals. It's a go-to technique when you encounter binomials multiplied by their conjugates. It simplifies the expression quickly and keeps your work organized.
Simplifying Expressions
Once the numerator has been rationalized using the conjugate and the difference of squares formula, the next step is to simplify the overall expression as much as possible. In our case, after rationalizing, the expression becomes: \(\frac{x-1}{x - 2\sqrt{x} + 1}\).
Simplification involves checking both the numerator and denominator for any like terms or possibilities to factor further. Although the numerator \(x - 1\) is straightforward, the denominator \(x - 2\sqrt{x} + 1\) may require additional attention. While it won't necessarily simplify into a neat whole number, ensuring you've captured each step correctly is key.
Always aim to express fractions in their simplest forms by cancelling out common factors and ensuring radicals are minimized effectively, supporting clearer and more manageable results.
Simplification involves checking both the numerator and denominator for any like terms or possibilities to factor further. Although the numerator \(x - 1\) is straightforward, the denominator \(x - 2\sqrt{x} + 1\) may require additional attention. While it won't necessarily simplify into a neat whole number, ensuring you've captured each step correctly is key.
Always aim to express fractions in their simplest forms by cancelling out common factors and ensuring radicals are minimized effectively, supporting clearer and more manageable results.
Other exercises in this chapter
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