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Chapter 0: Problem 10

In Exercises \(1-28\), compute the numbers. $$ (27)^{1 / 3} $$

Short Answer

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Step by step solution

01

Understand the Problem

The problem requires finding the cube root of the number 27. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.
02

Apply the Cube Root Function

The cube root of 27 is denoted as \(27^{1/3}\). To find \(27^{1/3}\), we need to determine what number multiplied by itself three times results in 27.
03

Determine the Cube Root Manually

To solve \(27^{1/3}\), consider the equation \(x^3 = 27\). Since \(3^3 = 27\), we have \(x = 3\). Thus, \(27^{1/3} = 3\).

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Key Concepts

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

What is a Cube Root?
To understand cube roots, let's break it down to basics. A cube root of a number is a value that, when multiplied by itself twice more (for a total of three times), results in the original number. In mathematical terms, the cube root of a number x is denoted as \(\sqrt[3]{x}\) or simply as \(x^{1/3}\). For example, the cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\).

  • Cube roots can be positive or negative. For example, the cube root of -8 is -2 because \((-2) \times (-2) \times (-2) = -8\).
  • Cubed numbers grow rapidly. For instance, \(10^3 = 1000\).
  • Unlike square roots, every real number has exactly one real cube root.
This concept is sometimes challenging, but grasping cube roots opens up further understanding of complex mathematical ideas.
Understanding Exponents
Exponents are a way to express repeated multiplication of the same number. The exponent denotes how many times the base number is multiplied by itself. For example, \(2^3\) means \(2 \times 2 \times 2\), which equals 8.

  • An essential rule is that any number to the power of 1 is itself. For example, \(5^1 = 5\).
  • Any number to the power of 0 is 1. For instance, \(7^0 = 1\).
  • Negative exponents represent division. For example, \(2^{-3} = 1 / (2^3) = 1/8\).
Combining exponents and roots, a cube root can be expressed as an exponent. A cube root of a number x, \(\sqrt[3]{x}\), can be written as \(x^{1/3}\). This dual representation helps in simplifying and solving various mathematical problems.
Roots in Mathematics
Roots play a vital role in mathematics and are the inverse operations of exponentiation. The square root of a number y, written as \(\sqrt{y}\), is a value that, when multiplied by itself, gives y. Similarly, the cube root, as introduced earlier, gives the value that, when multiplied by itself twice more, results in the original number.

  • The n-th root of a number x is written as \(\sqrt[n]{x}\) or \(x^{1/n}\). For example, the 4th root of 16 is written as \(\sqrt[4]{16}\) or \(16^{1/4}\), and it equals 2 because \(2 \times 2 \times 2 \times 2 = 16\).
  • Roots help in simplifying equations and understanding polynomial expressions.
  • Having a clear comprehension of roots is fundamental for higher-level mathematics, including calculus and algebra.
In conclusion, mastering roots, especially cube roots and their relationship with exponents, is essential for navigating through more complex mathematical problems.

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Most popular questions from this chapter

Let \(f(x)=x^{2}\). Graph the functions \(f(x)+1, f(x)-1, f(x)+2\), and \(f(x)-2 .\) Make a guess about the relationship between the graph of a general function \(f(x)\) and the graph of \(f(x)+c\) for some constant \(c\). Test your guess on the functions \(f(x)=x^{3}\) and \(f(x)=\sqrt{x}\).

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\left(-8 y^{9}\right)^{2 / 3}\)

The expressions in Exercises 83-88 may be factored as shown. Find the missing factors. \(2 x^{2 / 3}-x^{-1 / 3}=x^{-1 / 3}(\quad)\)

Convert the numbers from graphing calculator form to standard form (that is, without E). \(1.35 \mathrm{E} 13\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(x^{5} \cdot\left(\frac{y^{2}}{x}\right)^{3}\)
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