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Chapter 9: Problem 9

Find the \(n\)th root(s), if possible. The cube root of 27

Short Answer

Expert verified
The cube root of 27 is 3.

Step by step solution

01

Understanding the Cube Root

The cube root of a number is the value that, when multiplied by itself twice, gives the original number. It is denoted as \( \sqrt[3]{x} \) where x is the original number.
02

Find the Cube Root

The cube root of 27 can be written as \( \sqrt[3]{27} \). The whole numbers from 1 to 5 have cubes of 1, 8, 27, 64 and 125, respectively. As 27 is in the cube list, it is established that \( \sqrt[3]{27} = 3 \).

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Key Concepts

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

Understanding the Cube Root
A cube root is a fundamental concept in mathematics that deals with finding a number which, when multiplied by itself twice, equals a given value. This process is represented by the radical symbol with a small 3, as in \( \sqrt[3]{x} \). It is important to remember that:
  • Cube roots can apply to both positive and negative numbers. For example, the cube root of \(-8\) is \(-2\) because \((-2) \times (-2) \times (-2) = -8\).
  • Recognizing cubes of numbers can make calculating cube roots easier. For instance, knowing cubes like \(1, 8, 27, 64,\) etc., helps in quickly identifying cube roots.
In the given exercise, we look for the cube root of 27, which is an example of a perfect cube because there is an integer, 3, such that \(3 \times 3 \times 3 = 27\). Thus, the cube root of 27 is 3.
Algebra and Cube Roots
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It allows us to express general mathematical statements and solve equations effectively. Cube roots in algebra are especially useful because they can simplify equations and help us solve for unknowns when dealing with cubic expressions.
Applications of cube roots in algebra include:
  • Solving cubic equations of the form \(x^3 = a\), where finding the cube root of 'a' can give the value of 'x'.
  • Decomposing complex algebraic expressions and simplifying them by recognizing and applying cube root rules.
Moreover, understanding cube roots can assist in graphing cubic functions and analyzing their behavior, especially at points where the function intercepts a particular axis. This grasp of algebraic manipulation makes it easier to simplify complex mathematical problems.
Mathematical Operations with Cube Roots
Mathematical operations do not stop at addition, subtraction, multiplication, and division. Cube roots introduce another layer of complexity and fun to math. Here鈥檚 how cube roots fit into mathematical operations:
  • Multiplication: When dealing with cube roots, multiplying two cube roots together is equivalent to taking the cube root of the product of these numbers, e.g., \( \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b} \).
  • Division: Similarly, dividing cube roots equates to taking the cube root of the quotient, \( \sqrt[3]{a} \div \sqrt[3]{b} = \sqrt[3]{a \div b} \).
  • Combination with squares: Often, you might face expressions with both squares and cubes that need simplification. Understanding how to manage and simplify using both principles can be crucial in solving such problems.
Mastering these operations allows students to handle a broader range of mathematical problems, ensuring they feel comfortable tackling tasks that involve polynomials and radical expressions.

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Most popular questions from this chapter

Perform the indicated operation and simplify. $$\frac{5}{x+1} \cdot \frac{x+1}{25}$$

Rewrite the expression as a single fraction and simplify. $$3-\frac{1}{\sqrt{3}}$$

Find the distance between the two points. Round your answer to two decimal places, if necessary. $$(1,5),(2,-6)$$

Determine whether the lines are parallel, perpendicular, or neither. $$ \begin{aligned} &y=3 x+4 \\ &y=-3 x+4 \end{aligned} $$

Simplify the expression. $$(\sqrt[3]{2}-1)^{2}$$
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