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Chapter 9: Problem 71

Rationalize the denominator of the expression and simplify. $$\frac{9}{\sqrt{x}+2}$$

Short Answer

Expert verified
The rationalized and simplified form of the given expression is \(\frac{9\sqrt{x}-18}{x-4}\)

Step by step solution

01

Multiply by the Conjugate

Using the formula for the difference of squares, when two binomials that are the same except for the sign between terms are multiplied, the result is the square of the first term minus the square of the second term. Multiply the numerator and the denominator by the conjugate of the denominator to get: \((\frac{9}{\sqrt{x}+2}) * (\frac{\sqrt{x}-2}{\sqrt{x}-2})\)
02

Simplify the Fraction

Simplify the resulting expression by expanding and simplifying the numerator and the denominator: \(\frac{9\sqrt{x}-18}{x-4}\)
03

Further Simplification

This fraction cannot be simplified further as there are no common factors that can be canceled out from the numerator and the denominator. Hence, \(\frac{9\sqrt{x}-18}{x-4}\) is the simplified form of the given expression.

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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 essential when rationalizing denominators. It comes into play when you multiply two binomials that are identical except for the sign between them. This process will yield a result that's the difference between the square of the first term and the square of the second term. For instance, if you have
  • \((a+b)(a-b)\), it simplifies to \(a^2-b^2\).
In the context of our exercise where we have
  • \(\frac{9}{\sqrt{x}+2}\), multiplying the denominator by its conjugate, \(\sqrt{x}-2\), results in: \((\sqrt{x}+2)(\sqrt{x}-2) = x - 4\).
The denominator simplifies to \(x-4\), which simplifies rational expressions significantly.
Conjugate
The conjugate is a key tool when dealing with radical expressions in denominators. In mathematical terms, the conjugate of a binomial is created by changing the sign between two terms. When you have
  • \(\sqrt{x}+2\), the conjugate is \(\sqrt{x}-2\).
Using the conjugate is particularly useful because
  • multiplying a binomial by its conjugate facilitates the removal of roots from the denominator, making expressions simpler and more manageable.
For our particular problem, after multiplying the full fraction by the conjugate, we end up eliminating the square root from the denominator, leading to a simpler form.
Simplifying Expressions
Simplifying expressions involves performing operations to make an expression as concise as possible without changing its value. After rationalizing the original expression, our goal is to simplify the resulting fraction whenever possible.
  • So, we expand both the numerator and the denominator. The numerator becomes \(9(\sqrt{x} - 2)\), which expands to \(9\sqrt{x} - 18\).
  • The denominator, as mentioned earlier, simplifies to \(x - 4\).
In this case, no further simplification can be done because there are no common factors between the numerator and the denominator. Thus, the expression \(\frac{9\sqrt{x}-18}{x-4}\) is in its simplest form.
Binomials
Binomials, in mathematics, are algebraic expressions containing exactly two terms. They take the form of \(a+b\) or \(a-b\). In rationalizing denominators, binomials tend to occur both in the original problem and in the process of creating conjugates.
  • In our exercise, \(\sqrt{x}+2\) is the original binomial in the denominator.
  • The conjugate, \(\sqrt{x}-2\), is also a binomial.
Working with binomials involves understanding operations such as addition, subtraction, multiplication, and implementing distributive properties, especially in the context of binomials' interactions as formulas like the difference of squares.

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Most popular questions from this chapter

Simplify the expression. $$(\sqrt[5]{6}+3)^{2}$$

Simplify the expression. $$\sqrt{\frac{a}{4}}-\sqrt{\frac{a}{9}}$$

Airline Passengers An airline offers daily flights between Chicago and Denver. The total monthly cost \(C\) (in millions of dollars) of these flights is modeled by \(C=\sqrt{0.2 x+1}, x \geq 0\) where \(x\) is measured in thousands of passengers. The total cost of the flights for June is \(2.5\) million dollars. Approximately how many passengers flew in June?

Rationalize the denominator of the expression and simplify. (Assume all variables are positive.) $$\sqrt{\frac{7}{18}}$$

Reasoning Is \(x=25\) a solution of \(\sqrt{x}=-5\) ? Explain.
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