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Chapter 0: Problem 47

Rationalize the denominator. $$\frac{\sqrt{2}}{\sqrt{5}}$$

Short Answer

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

Step by step solution

01

Identify the denominator

In this exercise, the denominator of the fraction is \(\sqrt{5}\).
02

Multiply by the square root

To eliminate the square root in the denominator, multiply both the numerator and denominator by \(\sqrt{5}\). The fraction then becomes \(\frac{\sqrt{2} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}.\)
03

Simplify the fraction

Multiplying square roots together requires multiplying the numbers under the radical sign. This results in: \(\frac{\sqrt{2*5}}{5}\). Simplify further to \(\frac{\sqrt{10}}{5}\)

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Key Concepts

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

Simplifying Radicals
When we simplify radicals, our main goal is to make the radical expression as simple as possible while keeping it mathematically equivalent to its original form. Simplifying radical expressions involves reducing the expression to its simplest form by removing any perfect square factors from the radicand. The radicand is the number inside the square root symbol.

Here's how you simplify radicals:
  • Look for perfect square factors in the radicand (e.g., numbers like 4, 9, 16, etc.).
  • Extract these perfect squares from under the radical sign. For example, \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}.\)
  • Always ensure each factor under the radical sign is as minimal as possible.
Simplifying is a helpful skill, particularly when solving equations involving radicals or comparing different radical expressions.
Multiplying Square Roots
Multiplying square roots is a fundamental operation when dealing with problems involving radicals. The process is quite straightforward: you multiply the values under the radical signs to get a new radicand.

Here are the basic steps for multiplying square roots:
  • Take the square roots you are multiplying and set them under one radical sign. For example, to multiply \(\sqrt{a} \times \sqrt{b}\), you use the rule: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}.\)
  • Simplify the new radical, if possible, by applying radical simplification principles discussed earlier.
It's important to remember that this rule only holds for non-negative numbers because the square root of a negative number introduces complex numbers, which would not be covered in standard radical expressions.
Radical Expressions
Radical expressions are expressions that contain a square root, cube root, or any higher root. These can appear in different mathematical problems, and simplifying them or performing operations on them is a crucial skill.

Key points about radical expressions include:
  • Radicals are often written using the radical sign, \(\sqrt{}\), with the number or expression inside being the radicand.
  • Operations with radical expressions include addition, subtraction, multiplication, division, and even rationalization, which helps avoid square roots in denominators.
  • To "rationalize the denominator," we usually multiply both the numerator and denominator by a suitable radical to eliminate any radicals in the denominator.
  • Ensuring radicals are simplified completely is crucial in any arithmetic operation involving radicals.
Radical expressions may look complex, but understanding how to manipulate them makes solving many algebraic problems more accessible and manageable.

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Most popular questions from this chapter

What is a perfect square trinomial and how is it factored?

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with \(-3\) and ending with 3

Factor completely. $$x^{2 n}+6 x^{n}+8$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When checking a radical equation's proposed solution, I can substitute into the original equation or any equation that is part of the solution process.

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{4} \cdot \frac{8}{15}$$
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