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Chapter 7: Problem 19

Find each root. $$ \sqrt[3]{216} $$

Short Answer

Expert verified
The cube root of 216 is 6.

Step by step solution

01

Understand the Problem

The goal is to find the cube root of 216. This means finding a number which, when multiplied by itself three times, equals 216.
02

Recognize Perfect Cubes

To simplify finding the cube root, recognize that 216 is a perfect cube. A perfect cube is a number that can be written as the product of an integer times itself three times, i.e., \[ n^3 = 216 \].
03

Compute the Cube Root

We need to determine the integer that cubed gives 216. By examining the values and using knowledge of perfect cubes, we find that: \[6^3 = 6 \times 6 \times 6 = 216.\] Thus, \[ \sqrt[3]{216} = 6. \]

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Key Concepts

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

Perfect Cubes
To understand the concept of finding the cube root, you first need to know about perfect cubes. A perfect cube is a number that can be written as the product of an integer multiplied by itself three times. For example:
  • The number 8 is a perfect cube because \(2 \times 2 \times 2 = 8\).
  • The number 27 is a perfect cube because \(3 \times 3 \times 3 = 27\).
  • The number 64 is a perfect cube because \(4 \times 4 \times 4 = 64\).
  • We also have 216 as a perfect cube because \(6 \times 6 \times 6 = 216\).
Recognizing whether a number is a perfect cube helps simplify finding the cube root, as we need to identify the integer that, when cubed, results in the given number.
Cube Root
The cube root of a number is the value that, when multiplied by itself three times, gives the original number. In mathematical terms, if \(x^3 = n\), then \(\sqrt[3]{n} = x \). For example:
  • The cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
  • The cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\).
  • Hence, the cube root of 216 is 6 because \(6 \times 6 \times 6 = 216\).
This concept helps in determining cube roots quickly if you are familiar with perfect cubes. For \( \sqrt[3]{216} \), we see that 216 is a perfect cube, and the integer that, when cubed, equals 216, is 6. So, \(\sqrt[3]{216} = 6 \).
Integer Operations
Understanding integer operations is important when dealing with cube roots and perfect cubes. Multiplying, adding, and subtracting integers are basic arithmetic operations you use frequently.
When finding a cube root, you'll use multiplication extensively. For example, to verify that 6 is the cube root of 216, you should perform the following operation:
  • Multiply 6 by itself: \(6 \times 6 = 36 \).
  • Then, multiply the result by 6 again: \(36 \times 6 = 216 \).
Thus, understanding multiplication and how numbers interact helps you confirm that \(6\) is indeed the cube root of \(216\).Integer operations also play a role in general mathematics and help solve more complex problems as you progress.

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