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Chapter 29: Problem 2

\(\sqrt{2}+\sqrt{3}\)

Short Answer

Expert verified
The expression \(\sqrt{2} + \sqrt{3}\) cannot be simplified further and is approximately 3.146.

Step by step solution

01

Identify the Problem

We are given the expression \(\sqrt{2} + \sqrt{3}\), which involves two square root terms. Our task is to simplify or evaluate this expression.
02

Check for Simplification

Typically, when simplifying expressions involving square roots, we look for opportunities to combine terms. However, \(\sqrt{2}\) and \(\sqrt{3}\) are both irrational numbers and are not like terms. Thus, they cannot be combined further.
03

Approximate Each Term

To understand the sum better, we can approximate \(\sqrt{2}\) and \(\sqrt{3}\). The approximate value of \(\sqrt{2}\) is about 1.414 and that of \(\sqrt{3}\) is about 1.732. Adding these gives approximately 3.146.

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Key Concepts

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

Understanding Irrational Numbers
Irrational numbers are quite interesting and form an essential part of number systems in mathematics. They cannot be expressed as simple fractions or decimals that terminate or repeat.

Let's have a closer look at the nature of irrational numbers:
  • Irrational numbers include numbers like \(\sqrt{2}\), \(\sqrt{3}\), \(\pi\), and \(e\).
  • These numbers are infinite and non-repeating decimals.
  • Irrational numbers can be found on the number line, and they help us understand the spaces between whole numbers.
In essence, irrational numbers show the richness of the number line, filling in the gaps between the rational numbers.

When dealing with expressions like \(\sqrt{2} + \sqrt{3}\), recognizing these terms as irrational helps us understand why they don't simplify into neater fractions or whole numbers.
Exploring Square Roots
The square root is a fundamental mathematical operation that asks you to find what number multiplied by itself results in the original number.

Here is what you need to know about square roots:
  • The square root of a number \(x\) is commonly denoted as \(\sqrt{x}\).
  • Square roots have properties that help in various mathematics topics, such as the fact that \(\sqrt{x^2} = x\), assuming \(x\) is non-negative.
  • Many square roots result in irrational numbers unless the original number is a perfect square (like 1, 4, 9, etc.).
This helps explain why \(\sqrt{2}\) and \(\sqrt{3}\) appear in their radical forms rather than as simplified fractions. They represent two unique points within the number line that can't be further simplified in terms of basic arithmetic operations.
Approximating Radicals for Better Understanding
Approximating radicals can make irrational numbers more tangible, especially when we need a numerical answer rather than an exact radical form.

Here's how you can go about it:
  • Consider the square roots: to approximate \(\sqrt{2}\), you might know it equals about 1.414.
  • Similarly, \(\sqrt{3}\) approximates to about 1.732.
  • These decimal approximations allow for straightforward arithmetic operations, like addition or subtraction.
Using approximate values, \(\sqrt{2} + \sqrt{3}\) gives us an idea of its value, around 3.146. While approximations lack the precision of radicals, they offer practical insights for calculations in everyday life and allow us to grasp the relative scale of these numbers.

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Most popular questions from this chapter

In Exercises 9 through 16 , classify the given \(\alpha \in C\) as algebraic or transcendental over the given field \(F\). If \(\alpha\) is algebraic over \(F\), find \(\operatorname{deg}(\alpha, F)\). $$ \alpha=1+i, F=\mathbb{R} $$

Show that \(\left\\{a+b(\sqrt{2})+c(\sqrt[3]{2})^{2} \mid a, b, c \in Q\right\\}\) is a subfield of \(\mathbb{R}\) by using the ideas of this section, rather than by a formal verification of the field axioms. [Hint: Use Theorem 29.18.]

In Exercises 9 through 16 , classify the given \(\alpha \in C\) as algebraic or transcendental over the given field \(F\). If \(\alpha\) is algebraic over \(F\), find \(\operatorname{deg}(\alpha, F)\). $$ \left.\alpha=\sqrt{\pi}, F=\mathbb{Q}^{2} \pi\right) $$

Mark each of the following true or false. _____a. The number \(\pi\) is transcendental over \(Q\). _____b. \(\mathbb{C}\) is a simple extension of \(R\). _____c. Every element of a field \(F\) is algebraic over \(F\). _____d. \(\mathbb{R}\) is an extension field of \(Q\). _____e. \(Q\) is an extension field of \(\mathbb{Z}_{2}\). _____f. Let \(\alpha \in \mathbb{C}\) be algebraic over \(\mathbb{Q}\) of degree \(n\). If \(f(\alpha)=0\) for nonzero \(f(x) \in Q[x]\), then (degree \(f(x)) \geq n\). _____g. Let \(\alpha \in \mathbb{C}\) be algebraic over \(\mathbb{Q}\) of degree \(n\). If \(f(\alpha)=0\) for nonzero \(f(x) \in \mathbb{R}[x]\), then (degree \(f(x)) \geq n\). _____h. Every nonconstant polynomial in \(F[x]\) has a zero in some extension field of \(F\). _____I. Every nonconstant polynomial in \(F[x]\) has a zero in every extension field of \(F\). _____j. If \(x\) is an indeterminate, \(Q[\pi] \simeq Q[x]\).

\(\sqrt{3-\sqrt{6}}\)
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