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Chapter 5: Problem 70

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{x}}{\sqrt{x}-1}\)

Short Answer

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

Step by step solution

01

Identify the Conjugate of the Denominator

To rationalize the denominator, we need the conjugate of the denominator. If the denominator is \( \sqrt{x} - 1 \), its conjugate is \( \sqrt{x} + 1 \).
02

Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator of the fraction by \( \sqrt{x} + 1 \) to get: \[ \frac{\sqrt{x}}{\sqrt{x} - 1} \times \frac{\sqrt{x} + 1}{\sqrt{x} + 1} = \frac{\sqrt{x} (\sqrt{x} + 1)}{(\sqrt{x} - 1)(\sqrt{x} + 1)} \].
03

Simplify the Denominator using Difference of Squares

The denominator \((\sqrt{x} - 1)(\sqrt{x} + 1)\) can be simplified using the difference of squares formula \((a - b)(a + b) = a^2 - b^2\). So, \((\sqrt{x})^2 - 1^2 = x - 1\).
04

Distribute and Simplify the Numerator

Distribute \(\sqrt{x}\) in the numerator: \(\sqrt{x} \cdot \sqrt{x} + \sqrt{x} \cdot 1 = x + \sqrt{x}\).
05

Combine Simplified Numerator and Denominator

Now we can write the simplified expression: \[ \frac{x + \sqrt{x}}{x - 1} \]. This is the expression with a rationalized denominator.

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Key Concepts

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

Conjugate
When rationalizing a denominator, the first step often involves identifying the conjugate of the denominator. But, what exactly is a conjugate? In algebra, the conjugate of a binomial expression \(a - b\) is simply \(a + b\). In our exercise, the denominator \(\sqrt{x} - 1\) has the conjugate \(\sqrt{x} + 1\). The trick here is that multiplying a binomial by its conjugate enables the removal of radicals or complex numbers from the denominator.

Why use conjugates, you ask?
  • Conjugates create a "difference of squares" scenario.
  • They help simplify expressions by making the denominator a rational number.
If you find small radicals in denominators, try using conjugates to simplify your expressions effectively.
Difference of Squares
The concept of difference of squares is not as complicated as it sounds. It is simply an algebraic tool. When a binomial is multiplied by its conjugate, it leads to a difference of squares. The difference of squares formula is a useful math identity given by \(a^2 - b^2\) when \(a = b\).

For instance, multiplying \(\sqrt{x} - 1\) by \(\sqrt{x} + 1\) yields \( (\sqrt{x})^2 - 1^2 = x - 1\). Notice how this multiplication eliminates the radical that originally appeared in the denominator.

Using this formula effectively allows for radical expressions to be simplified, leading to a much more straightforward form.
Simplification
Simplification is a central component of algebra that makes mathematical expressions more manageable. Once you've rationalized the denominator using a conjugate, the next goal is to simplify the resulting expression.

In the context of the exercise, the numerator \(\sqrt{x}(\sqrt{x} + 1)\) simplifies to \(x + \sqrt{x}\). Here, you distribute \(\sqrt{x}\) by applying basic distributive law: multiply \(\sqrt{x}\) by each term in the binomial. Keep track of exponents while ensuring your work remains organized.
  • Combine like terms if possible.
  • Simplify expressions by eliminating unnecessary radicals.
Having a simplified expression can not only make further mathematical operations easier but also allows for better insights into what the expression represents.
Radicals
Radicals often present themselves as intimidating elements in mathematical expressions, but understanding them can demystify the process of rationalization. A radical usually indicates a root, such as a square root or cube root. In our exercise, \(\sqrt{x}\) appears as the square root, a common form of radicals.

Why are radicals important?
  • They help express numbers that are not perfect powers.
  • Allow for solving equations that involve roots.
While dealing with rationalizing denominators, addressing the radicals often involves methods like conjugates or pairing them in expressions to eliminate the root from the denominator altogether. Knowing how to handle radicals is crucial for success in higher mathematics and helps in general problem-solving scenarios.

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Most popular questions from this chapter

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). 117

Atomic masses are measured in atomic mass units (amu). The amu, \((1.66)\left(10^{-27}\right)\) kilograms, is defined as \(\frac{1}{12}\) the mass of a common carbon atom. Find the mass of a carbon atom in kilograms. Express the result in scientific notation.

Write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((8.76)(10)^{-5}\)

Use scientific notation and the properties of exponents to help you perform the following operations. \(\frac{0.000064}{16,000}\)

Write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((5.2)(10)^{-2}\)
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