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Chapter 0: Problem 47

In Exercises \(45-54,\) rationalize the denominator. $$\frac{\sqrt{2}}{\sqrt{5}}$$

Short Answer

Expert verified
The rationalized form of \(\frac{\sqrt{2}}{\sqrt{5}}\) is \(\frac{\sqrt{10}}{5}\).

Step by step solution

01

Identify the denominator

From the given fraction \(\frac{\sqrt{2}}{\sqrt{5}}\), it is clear that the denominator is \(\sqrt{5}\).
02

Rationalize the denominator

To rationalize the denominator, the numerator and the denominator needs to be multiplied by the root of the number in the denominator. Thus, the operation becomes: \(\frac{\sqrt{2}}{\sqrt{5}} * \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{10}}{5}\).
03

Check the Solved Result

After performing the operations in step 2, the denominator has become a rational number, which means it is successfully rationalized. The final result is: \(\frac{\sqrt{10}}{5}\)

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Key Concepts

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

Rational Expressions
Rational expressions are fractions where the numerator and/or the denominator is a polynomial. In these expressions, much like with numerical fractions, you need to consider operations such as addition, subtraction, multiplication, and division.
When dealing with rational expressions, always look for common factors in the numerator and denominator because these can help in simplifying the expression. However, unlike straightforward fractions, rational expressions can also involve polynomial division.
Here are some tips when working with rational expressions:
  • Factorize any polynomials where possible, to identify and cancel common factors.
  • Keep track of excluded values; these are values that make any denominator equal to zero.
  • Clear the denominators by multiplying every term by the least common denominator (LCD).
By mastering these steps, you'll find it easier to manipulate and simplify complex rational expressions.
Radical Expressions
Radical expressions are mathematical expressions that include a square root, cube root or higher-order roots. Understanding how to handle these can make complex expressions more manageable.
The key to working with radical expressions is simplification. This involves reducing the expression to its simplest form by applying several rules:
  • Product Rule: The square root of a product is equal to the product of the square roots of each factor. For example, \[ \sqrt{ab} = \sqrt{a} \times \sqrt{b} \]
  • Quotient Rule: This states that the square root of a quotient is equal to the quotient of the square roots. For example, \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]
  • Simplify radicals by factoring out perfect squares from under the square root.
These guidelines will help you approach radical expressions with confidence and ease.
Simplifying Fractions
Simplifying fractions, whether numerical or algebraic, involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1.
Simplified fractions are often easier to work with and can make solving equations and other calculations much simpler. For numerical fractions, this usually involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
For algebraic fractions, simplification requires:
  • Identifying common factors in both the numerator and denominator.
  • Cancelling out these common factors to reduce the fraction.
  • In complex expressions, combining like terms or using algebraic identities can also help.
Once you master simplifying fractions, you will find it less challenging to deal with more complex mathematical tasks.

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Most popular questions from this chapter

$$\begin{array}{l} \text { Find the exact value of } \sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}} \text { without } \\ \text { the use of a calculator. } \end{array}$$

Will help you prepare for the material covered in the first section of the next chapter. If \(y=|x+1|,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -4 and ending with 2

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathbf{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}-25}{x-5}=x-5$$
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