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Chapter 10: Problem 38

Rationalize each denominator. See Example 4. $$ \frac{-7}{\sqrt{x}-3} $$

Short Answer

Expert verified
\(-\frac{7\sqrt{x} + 21}{x - 9}\)

Step by step solution

01

Identify the Conjugate

The first step is to identify the conjugate of the denominator. The denominator is \( \sqrt{x} - 3 \), so its conjugate is \( \sqrt{x} + 3 \).
02

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \( \sqrt{x} + 3 \). This gives you:\[\frac{-7}{\sqrt{x} - 3} \times \frac{\sqrt{x} + 3}{\sqrt{x} + 3} = \frac{-7(\sqrt{x} + 3)}{(\sqrt{x} - 3)(\sqrt{x} + 3)}\]
03

Simplify the Denominator

The denominator is now the difference of squares. Simplify \((\sqrt{x} - 3)(\sqrt{x} + 3)\) using the identity \((a-b)(a+b) = a^2 - b^2\):\[(\sqrt{x})^2 - 3^2 = x - 9\]
04

Simplify the Numerator

Distribute the \(-7\) in the numerator:\[-7(\sqrt{x} + 3) = -7\sqrt{x} - 21\]
05

Combine and Simplify

Now combine the simplified numerator and denominator:\[\frac{-7\sqrt{x} - 21}{x - 9}\]This is the rationalized form of the original expression.

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Key Concepts

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

Understanding the Conjugate
When rationalizing the denominator, identifying the **conjugate** is a crucial step. The conjugate of a binomial expression like \(\sqrt{x} - 3\) is \(\sqrt{x} + 3\). This is derived by changing the sign between the terms in the expression.
  • In mathematics, the conjugate is used because it allows us to eliminate the square root in the denominator by creating a difference of squares.
  • Changing the sign is key鈥攊t pairs a sum with a difference, precisely transitioning to a more manageable form.
For instance, given the expression \(\frac{-7}{\sqrt{x} - 3}\), we use the conjugate \(\sqrt{x} + 3\) to simplify the denominator. By multiplying both the numerator and the denominator by this conjugate, we start the process of rationalization, which aims to cancel out any irrational terms in the denominator.
The Magic of Difference of Squares
The technique of using the **difference of squares** makes rationalizing much simpler and is essential when working with conjugates. The difference of squares formula is expressed as \((a-b)(a+b) = a^2 - b^2\). This identity is valuable because it quickly reduces complex expressions.
  • In our exercise, \((\sqrt{x} - 3)(\sqrt{x} + 3)\), becomes \(x - 9\).
  • This transformation works because squaring the square root \(\sqrt{x}\) yields \(x\), while squaring \(-3\) results in \(-9\).
This is a marvelous property because it swaps an irrational expression for a clean, simple polynomial. Recognizing and applying this pattern is fundamental whenever conjugates are in play, providing a clear pathway to simplifying complex expressions.
Simplifying Expressions Made Easy
The final step in rationalizing expressions is **simplifying expressions**, ensuring each component is presented in its simplest form.
  • In our solution, after multiplying the numerator, \(-7(\sqrt{x} + 3)\) becomes \(-7\sqrt{x} - 21\).
  • Combining this with the denominator, \(x - 9\), gives \(\frac{-7\sqrt{x} - 21}{x - 9}\).
Simplification often involves distribution, as seen when the constant \(-7\) was applied across \(\sqrt{x} + 3\). This step joins together all elements into a tidy expression. Always check if further reduction is possible. This ensures that the rationalized expression is as concise as possible, removing any lingering complexity, and making it easier to work with in subsequent calculations or problem-solving.

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Most popular questions from this chapter

Basal metabolic rate \((B M R)\) is the number of calories per day a person needs to maintain life. A person's basal metabolic rate \(B(w)\) in calories per day can be estimated with the function \(B(w)=70 w^{1 / 4},\) where \(w\) is the person's weight in kilograms. Use this information to answer Exercises. Estimate the BMR for a person who weighs 90 kilograms. Round to the nearest calorie. (Note: 90 kilograms is approximately 198 pounds.)

Use a calculator to write a four-decimal-place approximation of each number. $$ 18^{3 / 5} $$

Use rational expressions to write as a single radical expression. $$ \sqrt[5]{7} \cdot \sqrt[3]{y} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{x^{3}} $$

Use a calculator to write a four-decimal-place approximation of each number. $$ 76^{5 / 7} $$
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