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Chapter 5: Problem 7

Find the indicated real \(n\)th \(\operatorname{root}(\mathrm{s})\) of a. \(n=2, a=0\)

Short Answer

Expert verified
The square root of 0 is 0.

Step by step solution

01

Understand the problem

Given the exercise, we are tasked to find the value of the square root of 0. That means, we are looking for a number which when multiplied by itself gives 0.
02

Calculate the square root

Since any number multiplied by 0 equals 0, the square root of 0 is 0.

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Key Concepts

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

Understanding Square Roots
Square roots are an essential concept in mathematics. They represent a number which, when multiplied by itself, yields the original number. For example, when we say the square root of 9, we mean the number 3, since \[ 3 \times 3 = 9. \] Square roots are denoted by the symbol \(\sqrt{}\). When finding the square root of a number, especially a perfect square, the result is always a whole number. However, it is important to note:
  • Not all numbers have real square roots, especially when dealing with negative numbers.
  • When a negative number is under the square root, it typically results in an imaginary number (in real numbers, square roots are not negative).
This exercise involved finding the square root of 0. Since 0 multiplied by any number is 0, its square root is straightforwardly 0.
Mathematical Operations Involving Roots
Applying mathematical operations to roots involves recognizing each component's functions and results. When dealing with square roots, the operations reflect another layer of arithmetic rules:
  • The operation of taking a square root is the inverse of squaring a number.
  • Both addition and subtraction can distribute over square root expressions, namely, \(\sqrt{a} + \sqrt{b}\) is not equal to \(\sqrt{a+b}\).
  • The multiplication property of square roots states \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).
In our task, calculating \(\sqrt{0}\) requires the knowledge that a square root operation reduces to finding such a number which results in squaring zero to give zero.
Exploring Real Numbers
In mathematics, real numbers include all the numbers that can be found on the number line. This category encompasses both rational and irrational numbers. It is essential to comprehend their properties when working with roots:
  • Rational numbers are those that can be expressed as fractions, like 1/2 or 5, and include integers.
  • Irrational numbers, like \(\sqrt{2}\), cannot be written as simple fractions.
  • Real numbers are infinitely dense; between any two real numbers, another real number exists.
In the realm of real numbers, calculations, such as finding the square root, pertain to values existing not only as integers but also as decimals or fractions. In this context, when dealing with zeros, it's on the simpler end of this spectrum, as zero is a unique integer where its square root is consistent and clear.

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Most popular questions from this chapter

\(g(x)=\sqrt{x}-3\)

In Exercises 49鈥�52, determine whether the functions are inverses. $$ f(x)=\frac{x-3}{4}, g(x)=4 x+3 $$

\(h(x)=\sqrt{x+3}-3\)

\(f(x)=(6 x)^{1 / 2}+3\)

\(f(x)=\sqrt{x}\)
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