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Chapter 0: Problem 7

In Exercises determine whether the real number is rational or irrational. $$\sqrt[3]{64}$$

Short Answer

Expert verified
The number \( \sqrt[3]{64} \) is a rational number

Step by step solution

01

Understanding the terms

A rational number can be expressed as a fraction a/b where a and b are integers and b 鈮� 0. An irrational number cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal would go on forever, without repeating.
02

Find the cube root

The cube root of 64 is calculated by finding a number that when multiplied by itself three times equals 64. This yields the result is 4, because \(4 \times 4 \times 4 = 64\)
03

Determine if the number is rational or irrational

4 is a number that can be written as a fraction (4/1 for example), therefore making it a rational number

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Key Concepts

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

Cube Roots
The concept of cube roots revolves around finding a specific value that, when used as a factor three times, results in a given number. Unlike square roots, which involve only two factors, cube roots involve tripling the factor.
To calculate the cube root of a number like 64, we ask ourselves: "What number, when multiplied by itself twice more, equals 64?" The cube root of 64 is 4 because:
  • Multiplying 4 three times (i.e., 4 脳 4 脳 4) yields the result of 64.
  • Essentially, the cube root operation reverses raising a number to the power of three.
  • This can be generalized as: if \(b = \sqrt[3]{a} \), then \(b^3 = a\).
Cube roots are fundamental in mathematics, especially when dealing with volume-related problems where dimensions are involved.
Fraction Representation
Understanding fraction representation is key to distinguishing between rational and irrational numbers. A fraction consists of two parts:
  • A numerator (top number) representing the number of equal parts considered.
  • A denominator (bottom number) indicating the total number of equal parts that make up a whole.
For a number to be rational, it must be expressible as a fraction \(\frac{a}{b}\), where both "a" and "b" are integers and \(b eq 0\).
This expression signifies that the number is a ratio of two integers, hence the term "rational."
An example from our exercise is the number 4, which can be written as \(\frac{4}{1}\), fitting the definition of a rational number.
In contrast, irrational numbers, such as \(\sqrt{2}\) or \(\pi\), cannot be neatly expressed as fractions because their decimal expansions are infinite and non-repeating.
Mathematical Proofs
Mathematical proofs are logical sequences that establish the truth of a mathematical statement. They are essential for validating theories, formulas, and concepts within mathematics.
A proof involves a series of statements, each logically justified, that lead to a definitive conclusion.
  • There are various types of proofs, including direct proofs, indirect proofs, and proof by contradiction.
  • In direct proofs, statements are demonstrated through deductive reasoning from known facts and propositions.
  • For the exercise at hand, the reasoning proves the rationality of a number through evident fraction representation.
Proofs underpin the reliability and credibility of mathematical findings, ensuring that conclusions drawn are consistent and agreed upon within the mathematical community.
Such rigor helps solidify understanding and application of concepts like rational and irrational numbers across various scenarios.

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Most popular questions from this chapter

simplify each expression by factoring. $$ 2 x^{5 / 2}+x^{-1 / 2} $$

Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. $$ x^{3}-3 x^{2}-3 x-4 $$

evaluate the expression for the given value of x. $$ \sqrt[6]{x} \quad x=325 $$

evaluate the expression for the given value of x. $$ x^{-1 / 2} \quad x=4 $$

evaluate the expression for the given value of x. $$ \frac{1+x^{-1}}{x^{-1}} \quad x=3 $$
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